diff --git a/src/sage/rings/function_field/constructor.py b/src/sage/rings/function_field/constructor.py
index fa5a31cb683..9472e9baf5a 100644
--- a/src/sage/rings/function_field/constructor.py
+++ b/src/sage/rings/function_field/constructor.py
@@ -171,9 +171,9 @@ def create_key(self, polynomial, names):
         """
         if names is None:
             names = polynomial.variable_name()
-        if not isinstance(names,tuple):
+        if not isinstance(names, tuple):
             names = (names,)
-        return (polynomial,names,polynomial.base_ring())
+        return (polynomial, names, polynomial.base_ring())
 
     def create_object(self, version, key, **extra_args):
         """
@@ -197,7 +197,7 @@ def create_object(self, version, key, **extra_args):
         base_field = f.base_ring()
         if isinstance(base_field, function_field_rational.RationalFunctionField):
             k = base_field.constant_field()
-            if k.is_finite(): # then we are in positive characteristic
+            if k.is_finite():  # then we are in positive characteristic
                 # irreducible and separable
                 if f.is_irreducible() and not all(e % k.characteristic() == 0 for e in f.exponents()):
                     # monic and integral
diff --git a/src/sage/rings/function_field/derivations.py b/src/sage/rings/function_field/derivations.py
index 32252d41eff..dc5c91c9553 100644
--- a/src/sage/rings/function_field/derivations.py
+++ b/src/sage/rings/function_field/derivations.py
@@ -54,7 +54,7 @@ class FunctionFieldDerivation(RingDerivationWithoutTwist):
         sage: d
         d/dx
     """
-    def __init__(self, parent):
+    def __init__(self, parent) -> None:
         r"""
         Initialize a derivation.
 
diff --git a/src/sage/rings/function_field/derivations_polymod.py b/src/sage/rings/function_field/derivations_polymod.py
index 952d5813dd6..cf059ed8d53 100644
--- a/src/sage/rings/function_field/derivations_polymod.py
+++ b/src/sage/rings/function_field/derivations_polymod.py
@@ -39,7 +39,7 @@ class FunctionFieldDerivation_separable(FunctionFieldDerivation):
         sage: L.derivation()
         d/dx
     """
-    def __init__(self, parent, d):
+    def __init__(self, parent, d) -> None:
         """
         Initialize a derivation.
 
@@ -164,7 +164,7 @@ def _lmul_(self, factor):
 
 
 class FunctionFieldDerivation_inseparable(FunctionFieldDerivation):
-    def __init__(self, parent, u=None):
+    def __init__(self, parent, u=None) -> None:
         r"""
         Initialize this derivation.
 
@@ -286,7 +286,7 @@ class FunctionFieldHigherDerivation(Map):
           From: Rational function field in x over Finite Field of size 2
           To:   Rational function field in x over Finite Field of size 2
     """
-    def __init__(self, field):
+    def __init__(self, field) -> None:
         """
         Initialize.
 
@@ -386,7 +386,7 @@ class RationalFunctionFieldHigherDerivation_global(FunctionFieldHigherDerivation
         sage: h(x^2, 2)
         1
     """
-    def __init__(self, field):
+    def __init__(self, field) -> None:
         """
         Initialize.
 
@@ -580,7 +580,7 @@ class FunctionFieldHigherDerivation_global(FunctionFieldHigherDerivation):
         ((x^7 + 1)/x^2)*y^2 + x^3*y
     """
 
-    def __init__(self, field):
+    def __init__(self, field) -> None:
         """
         Initialize.
 
@@ -826,7 +826,7 @@ class FunctionFieldHigherDerivation_char_zero(FunctionFieldHigherDerivation):
         sage: h(h(h(e,1),1),1) == 3*2*h(e,3)
         True
     """
-    def __init__(self, field):
+    def __init__(self, field) -> None:
         """
         Initialize.
 
diff --git a/src/sage/rings/function_field/derivations_rational.py b/src/sage/rings/function_field/derivations_rational.py
index 777f16d3e08..73bb7965a70 100644
--- a/src/sage/rings/function_field/derivations_rational.py
+++ b/src/sage/rings/function_field/derivations_rational.py
@@ -31,7 +31,7 @@ class FunctionFieldDerivation_rational(FunctionFieldDerivation):
         sage: K.derivation()
         d/dx
     """
-    def __init__(self, parent, u=None):
+    def __init__(self, parent, u=None) -> None:
         """
         Initialize a derivation.
 
diff --git a/src/sage/rings/function_field/differential.py b/src/sage/rings/function_field/differential.py
index 8f841180532..d85e85ca105 100644
--- a/src/sage/rings/function_field/differential.py
+++ b/src/sage/rings/function_field/differential.py
@@ -98,7 +98,7 @@ class FunctionFieldDifferential(ModuleElement):
         sage: y.differential()                                                          # needs sage.rings.function_field
         ((21/4*x/(x^7 + 27/4))*y^2 + ((3/2*x^7 + 9/4)/(x^8 + 27/4*x))*y + 7/2*x^4/(x^7 + 27/4)) d(x)
     """
-    def __init__(self, parent, f, t=None):
+    def __init__(self, parent, f, t=None) -> None:
         """
         Initialize the differential `fdt`.
 
@@ -116,7 +116,7 @@ def __init__(self, parent, f, t=None):
 
         self._f = f
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of the differential.
 
@@ -132,7 +132,7 @@ def _repr_(self):
             sage: f.differential()
             (-1/x^2) d(x)
         """
-        if self._f.is_zero(): # zero differential
+        if self._f.is_zero():  # zero differential
             return '0'
 
         r = 'd({})'.format(self.parent()._gen_base_differential)
@@ -142,7 +142,7 @@ def _repr_(self):
 
         return '({})'.format(self._f) + ' ' + r
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a latex representation of the differential.
 
@@ -155,7 +155,7 @@ def _latex_(self):
             sage: latex(w)                                                              # needs sage.rings.function_field
             \left( x y^{2} + \frac{1}{x} y \right)\, dx
         """
-        if self._f.is_zero(): # zero differential
+        if self._f.is_zero():  # zero differential
             return '0'
 
         r = 'd{}'.format(self.parent()._gen_base_differential)
@@ -165,7 +165,7 @@ def _latex_(self):
 
         return '\\left(' + latex(self._f) + '\\right)\\,' + r
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash of ``self``.
 
@@ -396,7 +396,7 @@ def divisor(self):
         """
         F = self.parent().function_field()
         x = F.base_field().gen()
-        return self._f.divisor() + (-2)*F(x).divisor_of_poles() + F.different()
+        return self._f.divisor() + (-2) * F(x).divisor_of_poles() + F.different()
 
     def valuation(self, place):
         """
@@ -416,7 +416,7 @@ def valuation(self, place):
         """
         F = self.parent().function_field()
         x = F.base_field().gen()
-        return (self._f.valuation(place) + 2*min(F(x).valuation(place), 0)
+        return (self._f.valuation(place) + 2 * min(F(x).valuation(place), 0)
                 + F.different().valuation(place))
 
     def residue(self, place):
@@ -471,7 +471,7 @@ def residue(self, place):
             sage: sum([QQ(w.residue(p)) for p in d.support()])
             0
         """
-        R,fr_R,to_R = place._residue_field()
+        R, fr_R, to_R = place._residue_field()
 
         # Step 1: compute f such that fds equals this differential.
         s = place.local_uniformizer()
@@ -485,10 +485,10 @@ def residue(self, place):
             return R.zero()
         else:
             g_shifted = g * s**(-r)
-            c = g_shifted.higher_derivative(-r-1, s)
+            c = g_shifted.higher_derivative(-r - 1, s)
             return to_R(c)
 
-    def monomial_coefficients(self, copy=True):
+    def monomial_coefficients(self, copy: bool = True):
         """
         Return a dictionary whose keys are indices of basis elements in the
         support of ``self`` and whose values are the corresponding coefficients.
@@ -600,7 +600,7 @@ class DifferentialsSpace(UniqueRepresentation, Parent):
     """
     Element = FunctionFieldDifferential
 
-    def __init__(self, field, category=None):
+    def __init__(self, field, category=None) -> None:
         """
         Initialize the space of differentials of the function field.
 
@@ -627,7 +627,7 @@ def __init__(self, field, category=None):
         self._gen_base_differential = F.gen()
         self._gen_derivative_inv = ~der(F.gen())  # used for fast computation
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of the space of differentials.
 
@@ -770,7 +770,7 @@ class DifferentialsSpaceInclusion(Morphism):
           To:   Space of differentials of Function field in y defined by y^2 - x*y + 4*x^3
     """
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of this morphism.
 
@@ -790,7 +790,7 @@ def _repr_(self):
         s += "\n  To:   {}".format(self.codomain())
         return s
 
-    def is_injective(self):
+    def is_injective(self) -> bool:
         """
         Return ``True``, since the inclusion morphism is injective.
 
@@ -849,4 +849,4 @@ def _call_(self, v):
         """
         domain = self.domain()
         F = self.codomain().function_field()
-        return F(v._f)*F(domain._gen_base_differential).differential()
+        return F(v._f) * F(domain._gen_base_differential).differential()
diff --git a/src/sage/rings/function_field/divisor.py b/src/sage/rings/function_field/divisor.py
index 94f3ff2bce3..1cdd589dd58 100644
--- a/src/sage/rings/function_field/divisor.py
+++ b/src/sage/rings/function_field/divisor.py
@@ -147,7 +147,7 @@ class FunctionFieldDivisor(ModuleElement):
          + 3*Place (x, (1/(x^3 + x^2 + x))*y^2)
          - 6*Place (x + 1, y + 1)
     """
-    def __init__(self, parent, data):
+    def __init__(self, parent, data) -> None:
         """
         Initialize.
 
@@ -161,7 +161,7 @@ def __init__(self, parent, data):
         ModuleElement.__init__(self, parent)
         self._data = data
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash of the divisor.
 
@@ -214,8 +214,8 @@ def _format(self, formatter, mul, cr):
         if m == 1:
             r = formatter(p)
         elif m == -1:
-            r = '- ' + formatter(p) # seems more readable than `-`
-        else: # nonzero
+            r = '- ' + formatter(p)  # seems more readable than `-`
+        else:  # nonzero
             r = formatter(m) + mul + formatter(p)
         for p in places:
             m = self._data[p]
@@ -229,7 +229,7 @@ def _format(self, formatter, mul, cr):
                 r += cr + minus + formatter(-m) + mul + formatter(p)
         return r
 
-    def _repr_(self, split=True):
+    def _repr_(self, split: bool = True) -> str:
         """
         Return a string representation of the divisor.
 
@@ -249,7 +249,7 @@ def _repr_(self, split=True):
         """
         return self._format(repr, '*', '\n' if split else '')
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return the LaTeX representation of the divisor.
 
@@ -355,7 +355,7 @@ def _add_(self, other):
             True
         """
         divisor_group = self.parent()
-        places = set(self.support()).union( set(other.support()))
+        places = set(self.support()).union(set(other.support()))
         data = {}
         for place in places:
             m = self.multiplicity(place) + other.multiplicity(place)
@@ -583,7 +583,7 @@ def basis_function_space(self):
             sage: D.basis_function_space()
             [x/(x + 3), 1/(x + 3)]
         """
-        basis,_ = self._function_space()
+        basis, _ = self._function_space()
         return basis
 
     @cached_method
@@ -624,8 +624,8 @@ def to_V(f):
         from sage.rings.function_field.maps import (
             FunctionFieldLinearMap, FunctionFieldLinearMapSection)
 
-        mor_from_V = FunctionFieldLinearMap(Hom(V,F), from_V)
-        mor_to_V = FunctionFieldLinearMapSection(Hom(F,V), to_V)
+        mor_from_V = FunctionFieldLinearMap(Hom(V, F), from_V)
+        mor_to_V = FunctionFieldLinearMapSection(Hom(F, V), to_V)
 
         return V, mor_from_V, mor_to_V
 
@@ -728,8 +728,8 @@ def to_V(w):
         from sage.rings.function_field.maps import (
             FunctionFieldLinearMap, FunctionFieldLinearMapSection)
 
-        mor_from_V = FunctionFieldLinearMap(Hom(V,W), from_V)
-        mor_to_V = FunctionFieldLinearMapSection(Hom(W,V), to_V)
+        mor_from_V = FunctionFieldLinearMap(Hom(V, W), from_V)
+        mor_to_V = FunctionFieldLinearMapSection(Hom(W, V), to_V)
 
         return V, mor_from_V, mor_to_V
 
@@ -804,16 +804,16 @@ def _basis(self):
 
         # Step 2: construct matrix M of rational functions in x such that
         # M * B == C where B = [b1,b1,...,bn], C =[v1,v2,...,vn]
-        V,fr,to = F.free_module(map=True)
+        V, fr, to = F.free_module(map=True)
         B = matrix([to(b) for b in J.gens_over_base()])
         C = matrix([to(v) for v in I.gens_over_base()])
         M = C * B.inverse()
 
         # Step 2.5: get the denominator d of M and set mat = d * M
         den = lcm([e.denominator() for e in M.list()])
-        R = den.parent() # polynomial ring
+        R = den.parent()  # polynomial ring
         one = R.one()
-        mat = matrix(R, n, [e.numerator() for e in (den*M).list()])
+        mat = matrix(R, n, [e.numerator() for e in (den * M).list()])
         gens = list(I.gens_over_base())
 
         # Step 3: transform mat to a weak Popov form, together with gens
@@ -825,13 +825,13 @@ def _basis(self):
             bestp = -1
             best = -1
             for c in range(n):
-                d = mat[i,c].degree()
+                d = mat[i, c].degree()
                 if d >= best:
                     bestp = c
                     best = d
 
             if best >= 0:
-                pivot_row[bestp].append((i,best))
+                pivot_row[bestp].append((i, best))
                 if len(pivot_row[bestp]) > 1:
                     conflicts.append(bestp)
 
@@ -839,31 +839,31 @@ def _basis(self):
         while conflicts:
             c = conflicts.pop()
             row = pivot_row[c]
-            i,ideg = row.pop()
-            j,jdeg = row.pop()
+            i, ideg = row.pop()
+            j, jdeg = row.pop()
 
             if jdeg > ideg:
-                i,j = j,i
-                ideg,jdeg = jdeg,ideg
+                i, j = j, i
+                ideg, jdeg = jdeg, ideg
 
-            coeff = - mat[i,c].lc() / mat[j,c].lc()
+            coeff = - mat[i, c].lc() / mat[j, c].lc()
             s = coeff * one.shift(ideg - jdeg)
 
             mat.add_multiple_of_row(i, j, s)
             gens[i] += s * gens[j]
 
-            row.append((j,jdeg))
+            row.append((j, jdeg))
 
             bestp = -1
             best = -1
             for c in range(n):
-                d = mat[i,c].degree()
+                d = mat[i, c].degree()
                 if d >= best:
                     bestp = c
                     best = d
 
             if best >= 0:
-                pivot_row[bestp].append((i,best))
+                pivot_row[bestp].append((i, best))
                 if len(pivot_row[bestp]) > 1:
                     conflicts.append(bestp)
 
@@ -920,9 +920,9 @@ def pivot(v):
             for i in range(n):
                 e = v[i]
                 if e != 0:
-                    return (i,e.numerator().degree() - e.denominator().degree())
+                    return (i, e.numerator().degree() - e.denominator().degree())
 
-        def greater(v, w): # v and w are not equal
+        def greater(v, w):  # v and w are not equal
             return v[0] < w[0] or v[0] == w[0] and v[1] > w[1]
 
         # collate rows by their pivot position
@@ -942,7 +942,7 @@ def greater(v, w): # v and w are not equal
 
             head = pivots[0]
             for p in pivots[1:]:
-                if not greater(head,p):
+                if not greater(head, p):
                     head = p
 
             rows = pivot_rows[head]
@@ -953,7 +953,7 @@ def greater(v, w): # v and w are not equal
                 for i in rows[1:]:
                     vi = vbasis[i][head[0]]
                     ci = vi.numerator().lc() / vi.denominator().lc()
-                    vbasis[i] -= ci/cr * vbasis[r]
+                    vbasis[i] -= ci / cr * vbasis[r]
                     p = pivot(vbasis[i])
                     if p in pivot_rows:
                         pivot_rows[p].append(i)
@@ -968,11 +968,11 @@ def coordinates(f):
             coords = [k(0) for i in range(m)]
             while v != 0:
                 p = pivot(v)
-                ind = npivots.index(p) # an exception implies x is not in the domain
+                ind = npivots.index(p)  # an exception implies x is not in the domain
                 w = nbasis[ind]
                 cv = v[p[0]].numerator().lc() / v[p[0]].denominator().lc()
                 cw = w[p[0]].numerator().lc() / w[p[0]].denominator().lc()
-                c = cv/cw
+                c = cv / cw
                 v -= c * w
                 coords[ind] = c
             return coords
@@ -999,7 +999,7 @@ class DivisorGroup(UniqueRepresentation, Parent):
     """
     Element = FunctionFieldDivisor
 
-    def __init__(self, field):
+    def __init__(self, field) -> None:
         """
         Initialize.
 
@@ -1012,9 +1012,9 @@ def __init__(self, field):
         """
         Parent.__init__(self, base=IntegerRing(), category=CommutativeAdditiveGroups())
 
-        self._field = field # function field
+        self._field = field  # function field
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of the group of divisors.
 
@@ -1063,7 +1063,7 @@ def _coerce_map_from_(self, S):
         """
         if isinstance(S, PlaceSet):
             func = lambda place: prime_divisor(self._field, place)
-            return SetMorphism(Hom(S,self), func)
+            return SetMorphism(Hom(S, self), func)
 
     def function_field(self):
         """
@@ -1097,10 +1097,10 @@ def _an_element_(self):
         N = 10
         d = 1
         places = []
-        while len(places) <= N: # collect at least N places
+        while len(places) <= N:  # collect at least N places
             places += self._field.places(d)
             d += 1
         e = self.element_class(self, {})
-        for i in range(random.randint(0,N)):
+        for i in range(random.randint(0, N)):
             e += random.choice(places)
         return e
diff --git a/src/sage/rings/function_field/drinfeld_modules/action.py b/src/sage/rings/function_field/drinfeld_modules/action.py
index fb6ea191125..9bb50a00a18 100644
--- a/src/sage/rings/function_field/drinfeld_modules/action.py
+++ b/src/sage/rings/function_field/drinfeld_modules/action.py
@@ -81,7 +81,7 @@ class DrinfeldModuleAction(Action):
         True
     """
 
-    def __init__(self, drinfeld_module):
+    def __init__(self, drinfeld_module) -> None:
         """
         Initialize ``self``.
 
@@ -140,7 +140,7 @@ def _act_(self, pol, x):
             raise TypeError('second input must be in the field acted upon')
         return self._drinfeld_module(pol)(x)
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a LaTeX representation of the action.
 
@@ -160,7 +160,7 @@ def _latex_(self):
                f'{latex(self._base)}\\text{{{{ }}' \
                f'induced{{ }}by{{ }}}}{latex(self._drinfeld_module)}'
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of the action.
 
diff --git a/src/sage/rings/function_field/drinfeld_modules/charzero_drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/charzero_drinfeld_module.py
index ac3c1367cac..b9fc6416be8 100644
--- a/src/sage/rings/function_field/drinfeld_modules/charzero_drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/charzero_drinfeld_module.py
@@ -149,7 +149,7 @@ def _compute_coefficient_exp(self, k):
         c = self._base.zero()
         for i in range(k):
             j = k - i
-            c += self._compute_coefficient_exp(i)*self._compute_coefficient_log(j)**(q**i)
+            c += self._compute_coefficient_exp(i) * self._compute_coefficient_log(j)**(q**i)
         return -c
 
     def exponential(self, prec=Infinity, name='z'):
@@ -242,7 +242,7 @@ def coeff_exp(k):
             L = LazyPowerSeriesRing(self._base, name)
             return L(coeff_exp, valuation=1)
         L = PowerSeriesRing(self._base, name, default_prec=prec)
-        return L([0] + [coeff_exp(i) for i in range(1,prec)], prec=prec)
+        return L([0] + [coeff_exp(i) for i in range(1, prec)], prec=prec)
 
     @cached_method
     def _compute_coefficient_log(self, k):
@@ -274,8 +274,8 @@ def _compute_coefficient_log(self, k):
         for i in range(k):
             j = k - i
             if j < r + 1:
-                c += self._compute_coefficient_log(i)*self._gen[j]**(q**i)
-        return c/(T - T**(q**k))
+                c += self._compute_coefficient_log(i) * self._gen[j]**(q**i)
+        return c / (T - T**(q**k))
 
     def logarithm(self, prec=Infinity, name='z'):
         r"""
@@ -391,7 +391,7 @@ def _compute_goss_polynomial(self, n, q, poly_ring, X):
             pol += self._compute_coefficient_exp(i) * self._compute_goss_polynomial(n - m, q, poly_ring, X)
             m *= q
             i += 1
-        return X*(self._compute_goss_polynomial(n - 1, q, poly_ring, X) + pol)
+        return X * (self._compute_goss_polynomial(n - 1, q, poly_ring, X) + pol)
 
     def goss_polynomial(self, n, var='X'):
         r"""
@@ -600,7 +600,7 @@ def class_polynomial(self):
 
         # We compute the bound s
         gs = self.coefficients_in_function_ring(sparse=False)
-        s = max(gs[i].degree() // (q**i - 1) for i in range(1, r+1))
+        s = max(gs[i].degree() // (q**i - 1) for i in range(1, r + 1))
         if s == 0:
             return A.one()
 
@@ -609,9 +609,9 @@ def class_polynomial(self):
         # (for the standard structure of A-module!)
         M = matrix(Fq, s)
         qk = 1
-        for k in range(r+1):
+        for k in range(r + 1):
             for i in range(s):
-                e = (i+1)*qk
+                e = (i + 1) * qk
                 for j in range(s):
                     e -= 1
                     if e < 0:
@@ -626,10 +626,10 @@ def class_polynomial(self):
         # the Fq-vector space generated by the phi_T^i(T^(-s+1))
         # for i varying in NN.
         v = vector(Fq, s)
-        v[s-1] = 1
+        v[s - 1] = 1
         vs = [v]
-        for i in range(s-1):
-            v = v*M
+        for i in range(s - 1):
+            v = v * M
             vs.append(v)
         V = matrix(vs)
         V.echelonize()
@@ -643,7 +643,7 @@ def class_polynomial(self):
             while ip < dim and j == pivots[ip]:
                 j += 1
             ip += 1
-            V[i,j] = 1
+            V[i, j] = 1
         N = (V * M * ~V).submatrix(dim, dim)
 
         # The class module is now H where the action of T
diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index ed19c6064d9..b702997999e 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -621,7 +621,7 @@ def __classcall_private__(cls, function_ring, gen, name='τ'):
             return DrinfeldModule_charzero(gen, category)
         return cls.__classcall__(cls, gen, category)
 
-    def __init__(self, gen, category):
+    def __init__(self, gen, category) -> None:
         """
         Initialize ``self``.
 
@@ -750,7 +750,7 @@ def _Hom_(self, other, category):
         from sage.rings.function_field.drinfeld_modules.homset import DrinfeldModuleHomset
         return DrinfeldModuleHomset(self, other, category)
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a LaTeX representation of the Drinfeld module.
 
@@ -780,7 +780,7 @@ def _latex_(self):
             return f'\\phi: {latex(self._function_ring.gen())} \\mapsto ' \
                    f'{latex(self._gen)}'
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of this Drinfeld module.
 
@@ -797,7 +797,7 @@ def _repr_(self):
         return f'Drinfeld module defined by {self._function_ring.gen()} ' \
                f'|--> {self._gen}'
 
-    def _test_category(self, **options):
+    def _test_category(self, **options) -> None:
         """
         Run generic tests on the method :meth:`.category`.
 
@@ -824,7 +824,7 @@ def _test_category(self, **options):
         tester.assertTrue(isinstance(self, category.parent_class),
                 _LazyString("category of %s improperly initialized", (self,), {}))
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         r"""
         Return a hash of ``self``.
 
@@ -1000,7 +1000,7 @@ def basic_j_invariant_parameters(self, coeff_indices=None, nonzero=False):
                 raise TypeError('coefficients indices must be integers')
             if max(coeff_indices) >= r or min(coeff_indices) <= 0:
                 raise ValueError(f'indices must be > 0 and < {r}')
-            if not all(coeff_indices[i] < coeff_indices[i+1] for i in
+            if not all(coeff_indices[i] < coeff_indices[i + 1] for i in
                        range(len(coeff_indices) - 1)):
                 raise ValueError('indices must be distinct and sorted')
             if nonzero:
@@ -1022,7 +1022,7 @@ def basic_j_invariant_parameters(self, coeff_indices=None, nonzero=False):
             # Create inequalities of the form
             #   delta_i <= (q^r - 1)/(q^{gcd(i,r)} - 1)
             upper_bounds = [Integer((q**r - 1) / (q**(gcd(i, r)) - 1))]\
-                            + [0]*(len(coeff_indices) + 1)
+                            + [0] * (len(coeff_indices) + 1)
             upper_bounds[idx + 1] = -1
             inequalities.extend((lower_bounds, upper_bounds))
         equation.append(1 - q**r)
@@ -1270,7 +1270,7 @@ def height(self):
         except NotImplementedError:
             raise NotImplementedError('height not implemented in this case')
 
-    def is_isomorphic(self, other, absolutely=False):
+    def is_isomorphic(self, other, absolutely=False) -> bool:
         r"""
         Return ``True`` if this Drinfeld module is isomorphic to ``other``;
         return ``False`` otherwise.
@@ -1396,7 +1396,7 @@ def is_isomorphic(self, other, absolutely=False):
         B = other.gen()
         e = Integer(0)
         ue = self._base(1)
-        for i in range(1, r+1):
+        for i in range(1, r + 1):
             ai = A[i]
             bi = B[i]
             if ai == 0 and bi == 0:
@@ -1407,8 +1407,8 @@ def is_isomorphic(self, other, absolutely=False):
                 # u^e = ue
                 # u^(q^i - 1) = ai/bi
                 e, s, t = e.xgcd(q**i - 1)
-                ue = ue**s * (ai/bi)**t
-        for i in range(1, r+1):
+                ue = ue**s * (ai / bi)**t
+        for i in range(1, r + 1):
             if A[i]:
                 f = (q**i - 1) // e
                 if A[i] != B[i] * ue**f:
@@ -1666,14 +1666,14 @@ def j_invariant(self, parameter=None, check=True):
             if r != 2:
                 raise TypeError("parameter must not be None "
                                 "if the rank is greater than 2")
-            return self._gen[1]**(q+1)/self._gen[2]
+            return self._gen[1]**(q + 1) / self._gen[2]
         if parameter in ZZ:
             parameter = ZZ(parameter)
             if parameter <= 0 or parameter >= r:
                 raise ValueError("integer parameter must be >= 1 and < the "
                                  f"rank (={r})")
-            dk = Integer((q**r - 1)/(q**gcd(parameter, r) - 1))
-            dr = Integer((q**parameter - 1)/(q**gcd(parameter, r) - 1))
+            dk = Integer((q**r - 1) / (q**gcd(parameter, r) - 1))
+            dr = Integer((q**parameter - 1) / (q**gcd(parameter, r) - 1))
             return self._gen[parameter]**dk / self._gen[-1]**dr
         elif isinstance(parameter, (tuple, list)):
             if len(parameter) != 2:
@@ -1696,8 +1696,8 @@ def j_invariant(self, parameter=None, check=True):
             #   d_1 (q - 1) + ... + d_{r-1} (q^{r-1} - 1)
             #   = d_r (q^r - 1)
             if check:
-                right = parameter_1[-1]*(q**r - 1)
-                left = sum(parameter_1[i]*(q**(parameter_0[i]) - 1) for i in
+                right = parameter_1[-1] * (q**r - 1)
+                left = sum(parameter_1[i] * (q**(parameter_0[i]) - 1) for i in
                            range(len(parameter_0)))
                 if left != right:
                     raise ValueError("parameter does not satisfy the "
@@ -2120,4 +2120,4 @@ def frobenius_relative(self, n=1):
         d = self.characteristic().degree()
         if d < 0:
             raise ValueError("the characteristic of the Drinfeld module must be nonzero")
-        return self.hom(tau**(n*d))
+        return self.hom(tau**(n * d))
diff --git a/src/sage/rings/function_field/drinfeld_modules/finite_drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/finite_drinfeld_module.py
index c6a1f2ca08b..8a0f9234721 100644
--- a/src/sage/rings/function_field/drinfeld_modules/finite_drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/finite_drinfeld_module.py
@@ -124,7 +124,7 @@ class DrinfeldModule_finite(DrinfeldModule):
         True
     """
 
-    def __init__(self, gen, category):
+    def __init__(self, gen, category) -> None:
         """
         Initialize ``self``.
 
@@ -201,15 +201,15 @@ def _frobenius_matrix_crystalline(self):
         drin_coeffs = self.coefficients(sparse=False)
         poly_K = PolynomialRing(K, name=str(A.gen()))
         matrix_poly_K = MatrixSpace(poly_K, r, r)
-        mu_coeffs = ((poly_K.gen() - drin_coeffs[0])**(n+1)) \
+        mu_coeffs = ((poly_K.gen() - drin_coeffs[0])**(n + 1)) \
                     .coefficients(sparse=False)
 
         def companion(order):
             # + [1] is required to satisfy formatting for companion matrix
-            M = matrix_poly_K(companion_matrix([(drin_coeffs[i]/drin_coeffs[r])
-                               .frobenius(qdeg*order)
+            M = matrix_poly_K(companion_matrix([(drin_coeffs[i] / drin_coeffs[r])
+                               .frobenius(qdeg * order)
                                for i in range(r)] + [1], format='top'))
-            M[0, r-1] += poly_K.gen() / drin_coeffs[r].frobenius(qdeg*order)
+            M[0, r - 1] += poly_K.gen() / drin_coeffs[r].frobenius(qdeg * order)
             return M
 
         companion_initial = prod([companion(i) for i in range(nrem, 0, -1)])
@@ -218,12 +218,12 @@ def companion(order):
         reduced_companions = []
         for k in range(nquo - 1, 0, -1):
             M = Matrix(poly_K, r, r)
-            modulus = poly_K([c.frobenius(qdeg*(-k*nstar % n))
+            modulus = poly_K([c.frobenius(qdeg * (-k * nstar % n))
                                                 for c in mu_coeffs])
             for i, row in enumerate(companion_step):
                 for j, entry in enumerate(row):
                     reduction = entry % modulus
-                    M[i, j] = poly_K([c.frobenius(qdeg*(k*nstar))
+                    M[i, j] = poly_K([c.frobenius(qdeg * (k * nstar))
                                      for c in reduction
                                               .coefficients(sparse=False)])
             reduced_companions.append(M)
@@ -491,9 +491,9 @@ def _frobenius_charpoly_CSA(self):
         n = self._base_degree_over_constants
         r = self.rank()
         lc = chi[0][r]
-        coeffs = [A([K(chi[i][j]/lc).in_base()
-                     for i in range((r-j)*n // r + 1)])
-                  for j in range(r+1)]
+        coeffs = [A([K(chi[i][j] / lc).in_base()
+                     for i in range((r - j) * n // r + 1)])
+                  for j in range(r + 1)]
         return PolynomialRing(A, name='X')(coeffs)
 
     def _frobenius_charpoly_crystalline(self):
@@ -613,10 +613,10 @@ def _frobenius_charpoly_gekeler(self):
         # linear system. The system is prepared such that the solution
         # vector has the form [a_0, a_1, ... a_{r-1}]^T with each a_i
         # corresponding to a block of length (n*(r - i))//r + 1
-        shifts = [(n*(r - i))//r + 1 for i in range(r)]
-        rows, cols = n*r + 1, sum(shifts)
+        shifts = [(n * (r - i)) // r + 1 for i in range(r)]
+        rows, cols = n * r + 1, sum(shifts)
         block_shifts = [0]
-        for i in range(r-1):
+        for i in range(r - 1):
             block_shifts.append(block_shifts[-1] + shifts[i])
         # Compute the images \phi_T^i for i = 0 .. n.
         gen_powers = [self(A.gen()**i).coefficients(sparse=False)
@@ -626,7 +626,7 @@ def _frobenius_charpoly_gekeler(self):
         for j in range(r):
             for k in range(shifts[j]):
                 for i in range(len(gen_powers[k])):
-                    sys[i + n*j, block_shifts[j] + k] = gen_powers[k][i]
+                    sys[i + n * j, block_shifts[j] + k] = gen_powers[k][i]
         if sys.right_nullity() != 0:
             raise NotImplementedError("'gekeler' algorithm failed")
         sol = list(sys.solve_right(vec))
@@ -752,9 +752,9 @@ def frobenius_norm(self):
         r = self.rank()
         p = self.characteristic()
         norm = K(self.coefficients()[-1]).norm()
-        self._frobenius_norm = (-1) ** (n*r - n - r) \
+        self._frobenius_norm = (-1) ** (n * r - n - r) \
                                * norm**(-1) \
-                               * p ** (n//p.degree())
+                               * p ** (n // p.degree())
         return self._frobenius_norm
 
     def frobenius_trace(self, algorithm=None):
@@ -998,16 +998,16 @@ def invert(self, ore_pol):
         # Write the system and solve it
         k = deg // r
         A = self._function_ring
-        mat_rows = [[E.zero() for _ in range(k+1)] for _ in range(k+1)]
+        mat_rows = [[E.zero() for _ in range(k + 1)] for _ in range(k + 1)]
         mat_rows[0][0] = E.one()
         phiT = self.gen()
         phiTi = self.ore_polring().one()
-        for i in range(1, k+1):
+        for i in range(1, k + 1):
             phiTi *= phiT
-            for j in range(i+1):
-                mat_rows[j][i] = phiTi[r*j]
+            for j in range(i + 1):
+                mat_rows[j][i] = phiTi[r * j]
         mat = Matrix(mat_rows)
-        vec = vector([ore_pol[r*j] for j in range(k+1)])
+        vec = vector([ore_pol[r * j] for j in range(k + 1)])
         coeffs_K = list(mat.inverse() * vec)
         # Cast the coefficients to Fq
         try:
diff --git a/src/sage/rings/function_field/drinfeld_modules/homset.py b/src/sage/rings/function_field/drinfeld_modules/homset.py
index 4b5afb487c5..05e5cb17667 100644
--- a/src/sage/rings/function_field/drinfeld_modules/homset.py
+++ b/src/sage/rings/function_field/drinfeld_modules/homset.py
@@ -72,7 +72,7 @@ class DrinfeldModuleMorphismAction(Action):
           To:   Drinfeld module defined by T |--> (2*z^2 + 4*z + 4)*τ^2 + (z^2 + 4*z + 3)*τ + z
           Defn: τ + 2
     """
-    def __init__(self, A, H, is_left, op):
+    def __init__(self, A, H, is_left, op) -> None:
         r"""
         Initialize this action.
 
@@ -241,7 +241,7 @@ class DrinfeldModuleHomset(Homset):
     """
     Element = DrinfeldModuleMorphism
 
-    def __init__(self, X, Y, category=None, check=True):
+    def __init__(self, X, Y, category=None, check=True) -> None:
         """
         Initialize ``self``.
 
@@ -289,7 +289,7 @@ def __init__(self, X, Y, category=None, check=True):
             self.register_coercion(A)
         self._basis = None
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a LaTeX representation of the homset.
 
@@ -309,7 +309,7 @@ def _latex_(self):
                f'\\text{{{{ }}to{{ }}(gen){{ }}}}'\
                f'{latex(self.codomain().gen())}'
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of the homset.
 
@@ -327,7 +327,7 @@ def _repr_(self):
         return f'Set of Drinfeld module morphisms from (gen) '\
                f'{self.domain().gen()} to (gen) {self.codomain().gen()}'
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         r"""
         Return ``True`` if the input defines a morphism in the homset.
 
@@ -586,8 +586,8 @@ def _A_basis(self):
 
         # We compute the tau^i in M(phi)
         lc = ~(phiT[r])
-        xT = [-AF(lc*phiT[i]) for i in range(r)]
-        xT[0] += lc*TF
+        xT = [-AF(lc * phiT[i]) for i in range(r)]
+        xT[0] += lc * TF
         taus = []
         for i in range(r):
             taui = r * [AF.zero()]
@@ -595,14 +595,14 @@ def _A_basis(self):
             taus.append(taui)
         for i in range(r):
             s = FrobT(taui[-1])
-            taui = [s*xT[0]] + [FrobT(taui[j-1]) + s*xT[j] for j in range(1,r)]
+            taui = [s * xT[0]] + [FrobT(taui[j - 1]) + s * xT[j] for j in range(1, r)]
             taus.append(taui)
 
         # We precompute the Frob^k(z^i)
         z = F(Fo.gen())
         zs = []
         zq = z
-        for k in range(r+1):
+        for k in range(r + 1):
             x = F.one()
             for i in range(d):
                 zs.append(x)
@@ -617,17 +617,17 @@ def _A_basis(self):
                 #    sum(g_k*tau^k(x), k=0..r) - T*x
                 #  = sum(g_k*Frob^k(z^i)*tau^(k+j), k=0..r) - T*x
                 row = r * [AF.zero()]
-                for k in range(r+1):
-                    s = psiT[k] * zs[k*d + i]
+                for k in range(r + 1):
+                    s = psiT[k] * zs[k * d + i]
                     for l in range(r):
-                        row[l] += s*taus[k+j][l]
+                        row[l] += s * taus[k + j][l]
                 row[j] -= zs[i] * TF
                 # We write it in the A-basis
                 rowA = []
                 for c in row:
                     c0 = Fo(c[0]).vector()
                     c1 = Fo(c[1]).vector()
-                    rowA += [c0[k] + T*c1[k] for k in range(d)]
+                    rowA += [c0[k] + T * c1[k] for k in range(d)]
                 rows.append(rowA)
         M = Matrix(rows)
 
@@ -642,7 +642,7 @@ def _A_basis(self):
             u = S.zero()
             for i in range(d):
                 for j in range(r):
-                    a = ker[row, i*r + j]
+                    a = ker[row, i * r + j]
                     u += zs[i] * t**j * sum(a[k] * phiT**k for k in range(a.degree() + 1))
             isogenies.append(self(u))
 
@@ -711,32 +711,32 @@ def _Fq_basis(self, degree):
         # the Frobenius of K/Fq
         frob_matrices = [identity_matrix(Fq, n)] + [Matrix(Fq, n) for _ in range(d + r)]
         for i, elem in enumerate(K_basis):
-            for k in range(1, d+r+1):
+            for k in range(1, d + r + 1):
                 elem = elem ** q
                 v = elem.vector()
                 for j in range(n):
-                    frob_matrices[k][i,j] = v[j]
+                    frob_matrices[k][i, j] = v[j]
 
         # We write the linear system and solve it
-        sys = Matrix(Fq, (d + r + 1)*n, (d + 1)*n)
+        sys = Matrix(Fq, (d + r + 1) * n, (d + 1) * n)
         for k in range(0, d + r + 1):
             for i in range(max(0, k - r), min(k, d) + 1):
                 # We represent multiplication and Frobenius
                 # as operators acting on K as a vector space
                 # over Fq
-                oper = K(phiT[k-i] ** (q**i)).matrix() \
-                     - frob_matrices[k-i] * K(psiT[k-i]).matrix()
+                oper = K(phiT[k - i] ** (q**i)).matrix() \
+                     - frob_matrices[k - i] * K(psiT[k - i]).matrix()
                 for j in range(n):
                     for l in range(n):
-                        sys[k*n + j, i*n + l] = oper[l, j]
+                        sys[k * n + j, i * n + l] = oper[l, j]
         sol = sys.right_kernel().basis()
 
         # Reconstruct the Ore polynomial from the coefficients
         basis = []
         tau = domain.ore_polring().gen()
         for basis_elem in sol:
-            ore_poly = sum([sum([K_basis[j].backend() * basis_elem[i*n + j]
-                               for j in range(n)])*(tau**i)
+            ore_poly = sum([sum([K_basis[j].backend() * basis_elem[i * n + j]
+                               for j in range(n)]) * (tau**i)
                                for i in range(d + 1)])
             basis.append(self(ore_poly))
 
@@ -935,7 +935,7 @@ def basis_over_frobenius(self):
         # morphisms from domain to codomain.
         for j in range(n):
             for k in range(n):
-                for i in range(r+1):
+                for i in range(r + 1):
                     # Coefficients of tau^{i + k} coming from the
                     # relation defining morphisms of Drinfeld modules
                     # These are elements of K, expanded in terms of
@@ -943,8 +943,8 @@ def basis_over_frobenius(self):
                     poly = K(phiT[i]**(q**k) * K_basis[j]
                            - psiT[i] * K_basis[j]**(q**i)).polynomial()
                     deg = (i + k) // n
-                    row = n * (i + k - n*deg)
-                    col = k*n + j
+                    row = n * (i + k - n * deg)
+                    col = k * n + j
                     for b in range(poly.degree() + 1):
                         sys[row + b, col] += poly[b] * taun**deg
 
@@ -957,7 +957,7 @@ def basis_over_frobenius(self):
             basis_poly = 0
             for i in range(n):
                 for j in range(n):
-                    basis_poly += basis_vector[n*i + j].subs(tau**n) * K_basis[j].backend() * tau**i
+                    basis_poly += basis_vector[n * i + j].subs(tau**n) * K_basis[j].backend() * tau**i
             basis.append(self(basis_poly))
 
         return basis
diff --git a/src/sage/rings/function_field/drinfeld_modules/morphism.py b/src/sage/rings/function_field/drinfeld_modules/morphism.py
index 7a246fa8173..692ff9511d8 100644
--- a/src/sage/rings/function_field/drinfeld_modules/morphism.py
+++ b/src/sage/rings/function_field/drinfeld_modules/morphism.py
@@ -186,7 +186,7 @@ def __classcall_private__(cls, parent, x):
             raise ValueError('Ore polynomial does not define a morphism')
         return cls.__classcall__(cls, parent, ore_pol)
 
-    def __init__(self, parent, ore_pol):
+    def __init__(self, parent, ore_pol) -> None:
         r"""
         Initialize ``self``.
 
@@ -217,7 +217,7 @@ def __init__(self, parent, ore_pol):
         self._codomain = parent.codomain()
         self._ore_polynomial = ore_pol
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a LaTeX representation of the morphism.
 
@@ -235,7 +235,7 @@ def _latex_(self):
         """
         return f'{latex(self._ore_polynomial)}'
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of the morphism.
 
@@ -265,7 +265,7 @@ def _repr_(self):
                    f'  To:   {self._codomain}\n' \
                    f'  Defn: {self._ore_polynomial}'
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         r"""
         Return a hash of ``self``.
 
@@ -331,7 +331,7 @@ def is_identity(self):
         """
         return self._ore_polynomial == 1
 
-    def is_isogeny(self):
+    def is_isogeny(self) -> bool:
         r"""
         Return ``True`` whether the morphism is an isogeny.
 
@@ -695,7 +695,7 @@ def _motive_matrix(self):
         # The first row:
         # we write u = u0 + u1*phiT + u2*phiT^2 + ...
         u = self.ore_polynomial()
-        us = [ ]
+        us = []
         while not u.is_zero():
             u, ui = u.right_quo_rem(phiT)
             us.append(ui)
@@ -711,8 +711,8 @@ def _motive_matrix(self):
         T = KT.gen()
         for i in range(1, r):
             twist = [c.map_coefficients(Frob) for c in row]
-            row = [(inv*T - B[0]) * twist[-1]]
-            row += [twist[j-1] - B[j]*twist[-1] for j in range(1, r)]
+            row = [(inv * T - B[0]) * twist[-1]]
+            row += [twist[j - 1] - B[j] * twist[-1] for j in range(1, r)]
             rows.append(row)
 
         return matrix(KT, rows)
diff --git a/src/sage/rings/function_field/extensions.py b/src/sage/rings/function_field/extensions.py
index 94e810a9f87..212d1e6b884 100644
--- a/src/sage/rings/function_field/extensions.py
+++ b/src/sage/rings/function_field/extensions.py
@@ -83,7 +83,7 @@ class ConstantFieldExtension(FunctionFieldExtension):
 
     - ``k_ext`` -- an extension of `k`
     """
-    def __init__(self, F, k_ext):
+    def __init__(self, F, k_ext) -> None:
         """
         Initialize.
 
@@ -104,7 +104,7 @@ def __init__(self, F, k_ext):
             # construct constant field extension F_ext of F
             def_poly = F.polynomial().base_extend(F_ext_base)
             F_ext = F_ext_base.extension(def_poly, names=def_poly.variable_name())
-        else: # rational function field
+        else:  # rational function field
             F_ext = F_ext_base
 
         # embedding of F into F_ext
diff --git a/src/sage/rings/function_field/function_field.py b/src/sage/rings/function_field/function_field.py
index c46a231a21d..2e2ece950ad 100644
--- a/src/sage/rings/function_field/function_field.py
+++ b/src/sage/rings/function_field/function_field.py
@@ -234,15 +234,30 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
+from __future__ import annotations
+
+from typing import TYPE_CHECKING, Literal
+
+from sage.categories.function_fields import FunctionFields
+from sage.categories.homset import Hom
 from sage.misc.cachefunc import cached_method
 from sage.misc.lazy_import import LazyImport
+from sage.rings.integer import Integer
 from sage.rings.ring import Field
-from sage.categories.homset import Hom
-from sage.categories.function_fields import FunctionFields
 from sage.structure.category_object import CategoryObject
 
+if TYPE_CHECKING:
+    from sage.rings.function_field.divisor import DivisorGroup
+    from sage.rings.function_field.element import FunctionFieldElement
+    from sage.rings.function_field.extensions import ConstantFieldExtension
+    from sage.rings.function_field.function_field_rational import RationalFunctionField
+    from sage.rings.function_field.jacobian_base import Jacobian_base
+    from sage.rings.function_field.maps import FunctionFieldCompletion
+    from sage.rings.function_field.place import PlaceSet
+    from sage.rings.function_field.valuation import FunctionFieldValuation
 
-def is_FunctionField(x):
+
+def is_FunctionField(x) -> bool:
     """
     Return ``True`` if ``x`` is a function field.
 
@@ -284,7 +299,7 @@ class FunctionField(Field):
     """
     _differentials_space = LazyImport('sage.rings.function_field.differential', 'DifferentialsSpace')
 
-    def __init__(self, base_field, names, category=FunctionFields()):
+    def __init__(self, base_field, names, category=FunctionFields()) -> None:
         """
         Initialize.
 
@@ -304,7 +319,7 @@ def __init__(self, base_field, names, category=FunctionFields()):
         to_constant_base_field._make_weak_references()
         self.constant_base_field().register_conversion(to_constant_base_field)
 
-    def is_perfect(self):
+    def is_perfect(self) -> bool:
         r"""
         Return whether the field is perfect, i.e., its characteristic `p` is zero
         or every element has a `p`-th root.
@@ -318,7 +333,7 @@ def is_perfect(self):
         """
         return self.characteristic() == 0
 
-    def some_elements(self):
+    def some_elements(self) -> list[FunctionFieldElement]:
         """
         Return some elements in this function field.
 
@@ -361,13 +376,13 @@ def some_elements(self):
         for numerator in polynomials:
             for denominator in polynomials:
                 if denominator:
-                    some_element = numerator/denominator
+                    some_element = numerator / denominator
                     if some_element not in elements:
                         elements.append(some_element)
 
         return elements
 
-    def characteristic(self):
+    def characteristic(self) -> Integer:
         """
         Return the characteristic of the function field.
 
@@ -389,7 +404,7 @@ def characteristic(self):
         """
         return self.constant_base_field().characteristic()
 
-    def is_finite(self):
+    def is_finite(self) -> Literal[False]:
         """
         Return whether the function field is finite, which is false.
 
@@ -404,7 +419,7 @@ def is_finite(self):
         """
         return False
 
-    def is_global(self):
+    def is_global(self) -> bool:
         """
         Return whether the function field is global, that is, whether
         the constant field is finite.
@@ -456,7 +471,7 @@ def extension(self, f, names=None):
         from . import constructor
         return constructor.FunctionFieldExtension(f, names)
 
-    def order_with_basis(self, basis, check=True):
+    def order_with_basis(self, basis, check: bool = True):
         """
         Return the order with given basis over the maximal order of
         the base field.
@@ -504,7 +519,7 @@ def order_with_basis(self, basis, check=True):
         from .order_basis import FunctionFieldOrder_basis
         return FunctionFieldOrder_basis(tuple([self(a) for a in basis]), check=check)
 
-    def order(self, x, check=True):
+    def order(self, x, check: bool = True):
         """
         Return the order generated by ``x`` over the base maximal order.
 
@@ -541,13 +556,13 @@ def order(self, x, check=True):
         if len(x) == 1:
             g = x[0]
             basis = [self(1)]
-            for i in range(self.degree()-1):
-                basis.append(basis[-1]*g)
+            for i in range(self.degree() - 1):
+                basis.append(basis[-1] * g)
         else:
             raise NotImplementedError
         return self.order_with_basis(basis, check=check)
 
-    def order_infinite_with_basis(self, basis, check=True):
+    def order_infinite_with_basis(self, basis, check: bool = True):
         """
         Return the order with given basis over the maximal infinite order of
         the base field.
@@ -597,7 +612,7 @@ def order_infinite_with_basis(self, basis, check=True):
         from .order_basis import FunctionFieldOrderInfinite_basis
         return FunctionFieldOrderInfinite_basis(tuple([self(g) for g in basis]), check=check)
 
-    def order_infinite(self, x, check=True):
+    def order_infinite(self, x, check: bool = True):
         """
         Return the order generated by ``x`` over the maximal infinite order.
 
@@ -630,8 +645,8 @@ def order_infinite(self, x, check=True):
         if len(x) == 1:
             g = x[0]
             basis = [self(1)]
-            for i in range(self.degree()-1):
-                basis.append(basis[-1]*g)
+            for i in range(self.degree() - 1):
+                basis.append(basis[-1] * g)
         else:
             raise NotImplementedError
         return self.order_infinite_with_basis(tuple(basis), check=check)
@@ -729,7 +744,7 @@ def _coerce_map_from_(self, source):
                             # canonical, we require the names of the roots to match
                             return source.hom([sourcegen_in_self], base_morphism=base_coercion)
 
-    def _test_derivation(self, **options):
+    def _test_derivation(self, **options) -> None:
         """
         Test the correctness of the derivations of the function field.
 
@@ -754,13 +769,13 @@ def _test_derivation(self, **options):
             f = self.polynomial()
             tester.assertEqual(0, d(f))
         # Leibniz's law
-        for x,y in tester.some_elements(product(S, S)):
-            tester.assertEqual(d(x*y), x*d(y) + d(x)*y)
+        for x, y in tester.some_elements(product(S, S)):
+            tester.assertEqual(d(x * y), x * d(y) + d(x) * y)
         # Linearity
-        for x,y in tester.some_elements(product(S, S)):
-            tester.assertEqual(d(x+y), d(x) + d(y))
-        for c,x in tester.some_elements(product(K, S)):
-            tester.assertEqual(d(c*x), c*d(x))
+        for x, y in tester.some_elements(product(S, S)):
+            tester.assertEqual(d(x + y), d(x) + d(y))
+        for c, x in tester.some_elements(product(K, S)):
+            tester.assertEqual(d(c * x), c * d(x))
         # Constants map to zero
         for c in tester.some_elements(K):
             tester.assertEqual(d(c), 0)
@@ -847,7 +862,7 @@ def _intermediate_fields(self, base):
                 raise ValueError("field has not been constructed as a finite extension of base")
         return ret
 
-    def rational_function_field(self):
+    def rational_function_field(self) -> RationalFunctionField:
         r"""
         Return the rational function field from which this field has been
         created as an extension.
@@ -872,7 +887,7 @@ def rational_function_field(self):
 
         return self if isinstance(self, RationalFunctionField) else self.base_field().rational_function_field()
 
-    def valuation(self, prime):
+    def valuation(self, prime) -> FunctionFieldValuation:
         r"""
         Return the discrete valuation on this function field defined by
         ``prime``.
@@ -1067,7 +1082,7 @@ def basis_of_holomorphic_differentials(self):
 
     basis_of_differentials_of_first_kind = basis_of_holomorphic_differentials
 
-    def divisor_group(self):
+    def divisor_group(self) -> DivisorGroup:
         """
         Return the group of divisors attached to the function field.
 
@@ -1090,7 +1105,7 @@ def divisor_group(self):
         from .divisor import DivisorGroup
         return DivisorGroup(self)
 
-    def place_set(self):
+    def place_set(self) -> PlaceSet:
         """
         Return the set of all places of the function field.
 
@@ -1113,7 +1128,7 @@ def place_set(self):
         return PlaceSet(self)
 
     @cached_method
-    def completion(self, place, name=None, prec=None, gen_name=None):
+    def completion(self, place, name=None, prec=None, gen_name=None) -> FunctionFieldCompletion:
         """
         Return the completion of the function field at the place.
 
@@ -1216,7 +1231,7 @@ def completion(self, place, name=None, prec=None, gen_name=None):
         from .maps import FunctionFieldCompletion
         return FunctionFieldCompletion(self, place, name=name, prec=prec, gen_name=gen_name)
 
-    def hilbert_symbol(self, a, b, P):
+    def hilbert_symbol(self, a, b, P) -> Integer:
         r"""
         Return the Hilbert symbol `(a,b)_{F_P}` for the local field `F_P`.
 
@@ -1301,7 +1316,7 @@ def hilbert_symbol(self, a, b, P):
             raise ValueError('a and b must be elements of the function field')
 
         if a.is_zero() or b.is_zero():
-            return 0
+            return Integer(0)
 
         # Compute the completion map to precision 1 for computation of the
         # valuations v(a), v(b) as well as the elements a0, b0
@@ -1331,10 +1346,9 @@ def hilbert_symbol(self, a, b, P):
         # Finally, put the result together and transform it into the correct output
         res = k(-1)**(v_a * v_b * e) * a_rd_pw * b_rd_pw
 
-        from sage.rings.integer import Integer
         return Integer(1) if res.is_one() else Integer(-1)
 
-    def extension_constant_field(self, k):
+    def extension_constant_field(self, k) -> ConstantFieldExtension:
         """
         Return the constant field extension with constant field `k`.
 
@@ -1357,7 +1371,7 @@ def extension_constant_field(self, k):
         return ConstantFieldExtension(self, k)
 
     @cached_method
-    def jacobian(self, model=None, base_div=None, **kwds):
+    def jacobian(self, model=None, base_div=None, **kwds) -> Jacobian_base:
         """
         Return the Jacobian of the function field.
 
@@ -1415,9 +1429,8 @@ def jacobian(self, model=None, base_div=None, **kwds):
             if base_place is None:
                 raise ValueError('the function field has no rational place')
             # appropriate base divisor is constructed below.
-        else:
-            if isinstance(base_div, FunctionFieldPlace):
-                base_div = base_div.divisor()
+        elif isinstance(base_div, FunctionFieldPlace):
+            base_div = base_div.divisor()
 
         g = self.genus()
         curve = kwds.get('curve')
@@ -1426,15 +1439,15 @@ def jacobian(self, model=None, base_div=None, **kwds):
             from .jacobian_khuri_makdisi import Jacobian
             if model == 'km' or model.endswith('large'):
                 if base_div is None:
-                    base_div = (2*g + 1) * base_place
-                if not base_div.degree() >= 2*g + 1:
+                    base_div = (2 * g + 1) * base_place
+                if not base_div.degree() >= 2 * g + 1:
                     raise ValueError("Khuri-Makdisi large model requires base divisor of degree "
                                      "at least 2*g + 1 for genus g")
                 return Jacobian(self, base_div, model='large', curve=curve)
             elif model.endswith('medium'):
                 if base_div is None:
-                    base_div = (2*g + 1) * base_place
-                if not base_div.degree() >= 2*g + 1:
+                    base_div = (2 * g + 1) * base_place
+                if not base_div.degree() >= 2 * g + 1:
                     raise ValueError("Khuri-Makdisi medium model requires base divisor of degree "
                                      "at least 2*g + 1 for genus g")
                 return Jacobian(self, base_div, model='medium', curve=curve)
diff --git a/src/sage/rings/function_field/function_field_polymod.py b/src/sage/rings/function_field/function_field_polymod.py
index c158014d85c..cee41825462 100644
--- a/src/sage/rings/function_field/function_field_polymod.py
+++ b/src/sage/rings/function_field/function_field_polymod.py
@@ -26,6 +26,8 @@
 #                  http://www.gnu.org/licenses/
 # *****************************************************************************
 
+from typing import Literal
+
 from sage.arith.functions import lcm
 from sage.categories.function_fields import FunctionFields
 from sage.categories.homset import Hom
@@ -107,7 +109,7 @@ class FunctionField_polymod(FunctionField):
     """
     Element = FunctionFieldElement_polymod
 
-    def __init__(self, polynomial, names, category=None):
+    def __init__(self, polynomial, names, category=None) -> None:
         """
         Create a function field defined as an extension of another function
         field by adjoining a root of a univariate polynomial.
@@ -171,7 +173,7 @@ def __init__(self, polynomial, names, category=None):
         self._populate_coercion_lists_(coerce_list=[base_field, self._ring])
         self._gen = self(self._ring.gen())
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return hash of the function field.
 
@@ -186,7 +188,7 @@ def __hash__(self):
         """
         return self._hash
 
-    def _element_constructor_(self, x):
+    def _element_constructor_(self, x) -> FunctionFieldElement_polymod:
         r"""
         Make ``x`` into an element of the function field, possibly not canonically.
 
@@ -226,7 +228,7 @@ def gen(self, n=0):
             raise IndexError("there is only one generator")
         return self._gen
 
-    def ngens(self):
+    def ngens(self) -> Literal[1]:
         """
         Return the number of generators of the function field over its base
         field. This is by definition 1.
@@ -478,7 +480,7 @@ def _make_monic_integral(self, f):
         d = lcm([b.denominator() for b in f.list() if b])
         if d != 1:
             x = f.parent().gen()
-            g = (d**n) * f(x/d)
+            g = (d**n) * f(x / d)
         else:
             g = f
         return g, d
@@ -517,7 +519,7 @@ def constant_base_field(self):
         return self.base_field().constant_base_field()
 
     @cached_method(key=lambda self, base: self.base_field() if base is None else base)
-    def degree(self, base=None):
+    def degree(self, base=None) -> Integer:
         """
         Return the degree of the function field over the function field ``base``.
 
@@ -558,7 +560,7 @@ def degree(self, base=None):
             return ZZ(1)
         return self._polynomial.degree() * self.base_field().degree(base)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of the function field.
 
@@ -571,7 +573,7 @@ def _repr_(self):
         """
         return f"Function field in {self.variable_name()} defined by {self._polynomial}"
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return the LaTeX representation of the function field.
 
@@ -600,7 +602,7 @@ def base_field(self):
         """
         return self._base_field
 
-    def random_element(self, *args, **kwds):
+    def random_element(self, *args, **kwds) -> FunctionFieldElement_polymod:
         """
         Create a random element of the function field. Parameters are passed
         onto the random_element method of the base_field.
@@ -630,7 +632,7 @@ def polynomial(self):
         """
         return self._polynomial
 
-    def is_separable(self, base=None):
+    def is_separable(self, base=None) -> bool:
         r"""
         Return whether this is a separable extension of ``base``.
 
@@ -694,7 +696,7 @@ def polynomial_ring(self):
         return self._ring
 
     @cached_method(key=lambda self, base, basis, map: (self.base_field() if base is None else base, basis, map))
-    def free_module(self, base=None, basis=None, map=True):
+    def free_module(self, base=None, basis=None, map: bool = True):
         """
         Return a vector space and isomorphisms from the field to and from the
         vector space.
@@ -889,7 +891,7 @@ def equation_order(self):
             (1, x^3*y, x^6*y^2, x^9*y^3, x^12*y^4)
         """
         d = self._make_monic_integral(self.polynomial())[1]
-        return self.order(d*self.gen(), check=False)
+        return self.order(d * self.gen(), check=False)
 
     def hom(self, im_gens, base_morphism=None):
         """
@@ -1157,9 +1159,9 @@ def _simple_model(self, name='v'):
         factor = self.constant_base_field().zero()
         exponent = 0
         while True:
-            v = M(a+b*factor*x**exponent)
+            v = M(a + b * factor * x**exponent)
             minpoly = v.matrix(K).minpoly()
-            if minpoly.degree() == M.degree()*L.degree():
+            if minpoly.degree() == M.degree() * L.degree():
                 break
             factor += 1
             if factor == 0:
@@ -1542,7 +1544,7 @@ def separable_model(self, names=None):
             raise NotImplementedError("constructing a separable model is only implemented for function fields over a perfect constant base field")
 
         if names is None:
-            names = (self.variable_name()+"_", self.rational_function_field().variable_name()+"_")
+            names = (self.variable_name() + "_", self.rational_function_field().variable_name() + "_")
 
         L, from_L, to_L = self.monic_integral_model()
 
@@ -1720,7 +1722,7 @@ def _inversion_isomorphism(self):
         K = self.base_field()
         R = PolynomialRing(K, 'T')
         x = K.gen()
-        xinv = 1/x
+        xinv = 1 / x
 
         h = K.hom(xinv)
         F_poly = R([h(c) for c in self.polynomial().list()])
@@ -1731,7 +1733,7 @@ def _inversion_isomorphism(self):
 
         M, M2F, F2M = F.monic_integral_model('s')
 
-        return M, F2self*M2F, F2M*self2F
+        return M, F2self * M2F, F2M * self2F
 
     def places_above(self, p):
         """
@@ -1843,7 +1845,8 @@ def exact_constant_field(self, name='t'):
 
         return k_ext, embedding
 
-    def genus(self):
+    @cached_method
+    def genus(self) -> Integer:
         """
         Return the genus of the function field.
 
@@ -2042,7 +2045,7 @@ class FunctionField_global(FunctionField_simple):
     """
     _differentials_space = LazyImport('sage.rings.function_field.differential', 'DifferentialsSpace_global')
 
-    def __init__(self, polynomial, names):
+    def __init__(self, polynomial, names) -> None:
         """
         Initialize.
 
@@ -2283,7 +2286,7 @@ def _weierstrass_places(self):
         Mx = matrix(M.nrows(), [c._x for c in M.list()])
         detM = self(Mx.determinant() % self._polynomial)
 
-        R = detM.divisor() + sum(gaps)*W  # ramification divisor
+        R = detM.divisor() + sum(gaps) * W  # ramification divisor
 
         return R, gaps
 
@@ -2307,19 +2310,19 @@ def L_polynomial(self, name='t'):
         q = self.constant_field().order()
         g = self.genus()
 
-        B = [len(self.places(i+1)) for i in range(g)]
-        N = [sum(d * B[d-1] for d in ZZ(i+1).divisors()) for i in range(g)]
-        S = [N[i] - q**(i+1) - 1 for i in range(g)]
+        B = [len(self.places(i + 1)) for i in range(g)]
+        N = [sum(d * B[d - 1] for d in ZZ(i + 1).divisors()) for i in range(g)]
+        S = [N[i] - q**(i + 1) - 1 for i in range(g)]
 
         a = [1]
-        for i in range(1, g+1):
-            a.append(sum(S[j] * a[i-j-1] for j in range(i)) / i)
-        for j in range(1, g+1):
-            a.append(q**j * a[g-j])
+        for i in range(1, g + 1):
+            a.append(sum(S[j] * a[i - j - 1] for j in range(i)) / i)
+        for j in range(1, g + 1):
+            a.append(q**j * a[g - j])
 
         return ZZ[name](a)
 
-    def number_of_rational_places(self, r=1):
+    def number_of_rational_places(self, r=1) -> Integer:
         """
         Return the number of rational places of the function field whose
         constant field extended by degree ``r``.
@@ -2347,7 +2350,7 @@ def number_of_rational_places(self, r=1):
         R = IntegerRing()[[L.parent().gen()]]  # power series ring
 
         f = R(Lp / L, prec=r)
-        n = f[r-1] + q**r + 1
+        n = f[r - 1] + q**r + 1
 
         return n
 
@@ -2484,7 +2487,7 @@ def _maximal_order_basis(self):
         for f in pols_in_S:
             p = f.polynomial(S.gen(0))
             s = 0
-            for i in range(p.degree()+1):
+            for i in range(p.degree() + 1):
                 s += p[i].subs(x) * y**i
             pols.append(s)
 
@@ -2511,7 +2514,7 @@ def _maximal_order_basis(self):
         basis_V = [to_V(bvec) for bvec in _basis]
         l = lcm([vvec.denominator() for vvec in basis_V])
 
-        _mat = matrix([[coeff.numerator() for coeff in l*v] for v in basis_V])
+        _mat = matrix([[coeff.numerator() for coeff in l * v] for v in basis_V])
         reversed_hermite_form(_mat)
 
         basis = [fr_V(v) / l for v in _mat if not v.is_zero()]
@@ -2563,9 +2566,9 @@ def primitive_integal_element_infinite(self):
         y = self.gen()
         x = self.base_field().gen()
 
-        cf = max([(f[i].numerator().degree()/(n-i)).ceil() for i in range(n)
+        cf = max([(f[i].numerator().degree() / (n - i)).ceil() for i in range(n)
                   if f[i] != 0])
-        return y*x**(-cf)
+        return y * x**(-cf)
 
     @cached_method
     def equation_order_infinite(self):
diff --git a/src/sage/rings/function_field/function_field_rational.py b/src/sage/rings/function_field/function_field_rational.py
index 74ebb736992..ffc72b51f33 100644
--- a/src/sage/rings/function_field/function_field_rational.py
+++ b/src/sage/rings/function_field/function_field_rational.py
@@ -25,6 +25,8 @@
 #                  http://www.gnu.org/licenses/
 # *****************************************************************************
 
+from typing import Literal
+
 from sage.arith.functions import lcm
 from sage.categories.function_fields import FunctionFields
 from sage.categories.homset import Hom
@@ -125,7 +127,7 @@ class RationalFunctionField(FunctionField):
     """
     Element = FunctionFieldElement_rational
 
-    def __init__(self, constant_field, names, category=None):
+    def __init__(self, constant_field, names, category=None) -> None:
         """
         Initialize.
 
@@ -191,7 +193,7 @@ def __reduce__(self):
         from .constructor import FunctionField
         return FunctionField, (self._constant_field, self._names)
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return hash of the function field.
 
@@ -205,7 +207,7 @@ def __hash__(self):
         """
         return self._hash
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return string representation of the function field.
 
@@ -218,7 +220,7 @@ def _repr_(self):
         return "Rational function field in %s over %s" % (
             self.variable_name(), self._constant_field)
 
-    def _element_constructor_(self, x):
+    def _element_constructor_(self, x) -> FunctionFieldElement_rational:
         r"""
         Coerce ``x`` into an element of the function field, possibly not canonically.
 
@@ -480,7 +482,7 @@ def polynomial_ring(self, var='x'):
         return self[var]
 
     @cached_method(key=lambda self, base, basis, map: map)
-    def free_module(self, base=None, basis=None, map=True):
+    def free_module(self, base=None, basis=None, map: bool = True):
         """
         Return a vector space `V` and isomorphisms from the field to `V` and
         from `V` to the field.
@@ -545,7 +547,7 @@ def free_module(self, base=None, basis=None, map=True):
         to_V = MapFunctionFieldToVectorSpace(self, V)
         return (V, from_V, to_V)
 
-    def random_element(self, *args, **kwds):
+    def random_element(self, *args, **kwds) -> FunctionFieldElement_rational:
         """
         Create a random element of the rational function field.
 
@@ -559,7 +561,7 @@ def random_element(self, *args, **kwds):
         """
         return self(self._field.random_element(*args, **kwds))
 
-    def degree(self, base=None):
+    def degree(self, base=None) -> Integer:
         """
         Return the degree over the base field of the rational function
         field.  Since the base field is the rational function field itself, the
@@ -603,7 +605,7 @@ def gen(self, n=0):
             raise IndexError("Only one generator.")
         return self._gen
 
-    def ngens(self):
+    def ngens(self) -> Literal[1]:
         """
         Return the number of generators, which is 1.
 
@@ -770,7 +772,7 @@ def different(self):
         """
         return self.divisor_group().zero()
 
-    def genus(self):
+    def genus(self) -> Integer:
         """
         Return the genus of the function field, namely 0.
 
diff --git a/src/sage/rings/function_field/function_field_test.py b/src/sage/rings/function_field/function_field_test.py
index e8a1359eadf..a5822679c87 100644
--- a/src/sage/rings/function_field/function_field_test.py
+++ b/src/sage/rings/function_field/function_field_test.py
@@ -18,23 +18,23 @@ def J():
 
 @pytest.fixture
 def K():
-    return FunctionField(FiniteField(5**2,'a'), 'x')
+    return FunctionField(FiniteField(5**2, 'a'), 'x')
 
 
 @pytest.fixture
 def L(F):
     x = F.gen()
     Y = PolynomialRing(F, 'Y').gen()
-    return F.extension(Y**2 + Y + x + 1/x, 'y')
+    return F.extension(Y**2 + Y + x + 1 / x, 'y')
 
 
 @pytest.fixture
 def M(K, R, S):
     x = K.gen()
     y = R.gen()
-    L = K.extension(y**3 - (x**3 + 2*x*y + 1/x))
+    L = K.extension(y**3 - (x**3 + 2 * x * y + 1 / x))
     t = S.gen()
-    return L.extension(t**2 - x*y)
+    return L.extension(t**2 - x * y)
 
 
 @pytest.fixture
@@ -56,7 +56,7 @@ def R(K):
 def S(K, R):
     x = K.gen()
     y = R.gen()
-    L = K.extension(y**3 - (x**3 + 2*x*y + 1/x))
+    L = K.extension(y**3 - (x**3 + 2 * x * y + 1 / x))
     return PolynomialRing(L, 't')
 
 
@@ -72,18 +72,18 @@ def T(F):
 #
 #   https://github.com/pytest-dev/pytest/issues/349
 #
-pairs = [ ("J", None),
+pairs = [("J", None),
           ("K", 16),
           ("L", 2),
           ("M", 1),
           ("N", 1),
           ("O", None),
           ("T", None),
-          ("S", 8) ]
+          ("S", 8)]
 
 
 @pytest.mark.parametrize("ff,max_runs", pairs)
-def test_function_field_testsuite(ff, max_runs, request):
+def test_function_field_testsuite(ff, max_runs, request) -> None:
     r"""
     Run the TestSuite() on some function fields that are
     constructed in the documentation. They are slow, random, and not
@@ -109,7 +109,7 @@ def test_function_field_testsuite(ff, max_runs, request):
     ff = request.getfixturevalue(ff)
 
     # Pass max_runs only if it's not None; otherwise use the default
-    run_args = { "verbose": True, "raise_on_failure": True }
+    run_args = {"verbose": True, "raise_on_failure": True}
     if max_runs:
         run_args["max_runs"] = max_runs
 
diff --git a/src/sage/rings/function_field/ideal.py b/src/sage/rings/function_field/ideal.py
index e00bae907e4..2daa6e9ecbf 100644
--- a/src/sage/rings/function_field/ideal.py
+++ b/src/sage/rings/function_field/ideal.py
@@ -119,7 +119,7 @@ class FunctionFieldIdeal(Element):
         sage: O.ideal(x^3 + 1)
         Ideal (x^3 + 1) of Maximal order of Rational function field in x over Finite Field of size 7
     """
-    def __init__(self, ring):
+    def __init__(self, ring) -> None:
         """
         Initialize.
 
@@ -134,7 +134,7 @@ def __init__(self, ring):
         Element.__init__(self, ring.ideal_monoid())
         self._ring = ring
 
-    def _repr_short(self):
+    def _repr_short(self) -> str:
         """
         Return a string representation of this ideal that doesn't
         include the name of the ambient ring.
@@ -154,7 +154,7 @@ def _repr_short(self):
 
         return "(%s)" % (', '.join([repr(g) for g in self.gens_reduced()]), )
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return a string representation of this ideal.
 
@@ -214,7 +214,7 @@ def _repr_(self):
 
         return "Ideal %s of %s" % (self._repr_short(), self.ring())
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return the LaTeX representation of the ideal.
 
@@ -607,7 +607,7 @@ class FunctionFieldIdeal_module(FunctionFieldIdeal, Ideal_generic):
         sage: I^2
         Ideal (x^3 + 1, (-x^3 - 1)*y) of Order in Function field in y defined by y^2 - x^3 - 1
     """
-    def __init__(self, ring, module):
+    def __init__(self, ring, module) -> None:
         """
         Initialize.
 
@@ -631,7 +631,7 @@ def __init__(self, ring, module):
         # module generators are still ideal generators
         Ideal_generic.__init__(self, ring, self._gens, coerce=False)
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         Return ``True`` if ``x`` is in this ideal.
 
@@ -652,7 +652,7 @@ def __contains__(self, x):
         """
         return self._structure[2](x) in self._module
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash of this ideal.
 
@@ -665,7 +665,7 @@ def __hash__(self):
             sage: I = O.ideal(y)
             sage: d = {I: 1}  # indirect doctest
         """
-        return hash((self._ring,self._module))
+        return hash((self._ring, self._module))
 
     def _richcmp_(self, other, op):
         """
@@ -803,7 +803,7 @@ def _mul_(self, other):
             sage: I * J
             Ideal ((-x^5 + x^4 - x^2 + x)*y + x^3 + 1, (x^3 - x^2 + 1)*y) of Order in Function field in y defined by y^2 - x^3 - 1
         """
-        return self.ring().ideal([x*y for x in self.gens() for y in other.gens()])
+        return self.ring().ideal([x * y for x in self.gens() for y in other.gens()])
 
     def _acted_upon_(self, other, on_left):
         """
@@ -843,7 +843,7 @@ def intersection(self, other):
             try:
                 if self.ring().has_coerce_map_from(other):
                     return self
-            except (TypeError,ArithmeticError,ValueError):
+            except (TypeError, ArithmeticError, ValueError):
                 pass
             other = self.ring().ideal(other)
 
@@ -912,7 +912,7 @@ class FunctionFieldIdealInfinite_module(FunctionFieldIdealInfinite, Ideal_generi
         sage: O.ideal(y)
         Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1
     """
-    def __init__(self, ring, module):
+    def __init__(self, ring, module) -> None:
         """
         Initialize.
 
@@ -937,7 +937,7 @@ def __init__(self, ring, module):
         # module generators are still ideal generators
         Ideal_generic.__init__(self, ring, self._gens, coerce=False)
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         Return ``True`` if ``x`` is in this ideal.
 
@@ -962,7 +962,7 @@ def __contains__(self, x):
         """
         return self._structure[2](x) in self._module
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash of this ideal.
 
@@ -975,7 +975,7 @@ def __hash__(self):
             sage: I = O.ideal_with_gens_over_base([1, y])
             sage: d = {I: 2}  # indirect doctest
         """
-        return hash((self._ring,self._module))
+        return hash((self._ring, self._module))
 
     def __eq__(self, other):
         """
@@ -1052,7 +1052,7 @@ class IdealMonoid(UniqueRepresentation, Parent):
         Monoid of ideals of Maximal order of Rational function field in x over Finite Field of size 2
     """
 
-    def __init__(self, R):
+    def __init__(self, R) -> None:
         """
         Initialize the ideal monoid.
 
@@ -1069,7 +1069,7 @@ def __init__(self, R):
         self.__R = R
         self._populate_coercion_lists_()
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of the ideal monoid.
 
@@ -1109,7 +1109,7 @@ def _element_constructor_(self, x):
             sage: M([x-4, 1/x])
             Ideal (1/x) of Maximal order of Rational function field in x over Finite Field of size 2
         """
-        try: # x is an ideal
+        try:  # x is an ideal
             x = x.gens()
         except AttributeError:
             pass
diff --git a/src/sage/rings/function_field/ideal_polymod.py b/src/sage/rings/function_field/ideal_polymod.py
index da0c837e963..d7b6f22eede 100644
--- a/src/sage/rings/function_field/ideal_polymod.py
+++ b/src/sage/rings/function_field/ideal_polymod.py
@@ -52,7 +52,7 @@ class FunctionFieldIdeal_polymod(FunctionFieldIdeal):
         sage: O.ideal(y)
         Ideal (y) of Maximal order of Function field in y defined by y^2 + x^3*y + x
     """
-    def __init__(self, ring, hnf, denominator=1):
+    def __init__(self, ring, hnf, denominator=1) -> None:
         """
         Initialize.
 
@@ -93,7 +93,7 @@ def __init__(self, ring, hnf, denominator=1):
         # if the second gen is zero, the tuple has only the first gen.
         self._gens_two_vecs = None
 
-    def __bool__(self):
+    def __bool__(self) -> bool:
         """
         Test if this ideal is zero.
 
@@ -128,7 +128,7 @@ def __bool__(self):
         """
         return self._hnf.nrows() != 0
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash of this ideal.
 
@@ -151,7 +151,7 @@ def __hash__(self):
         """
         return hash((self._ring, self._hnf, self._denominator))
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         Return ``True`` if ``x`` is in this ideal.
 
@@ -833,7 +833,7 @@ def norm(self):
         return n
 
     @cached_method
-    def is_prime(self):
+    def is_prime(self) -> bool:
         """
         Return ``True`` if this ideal is a prime ideal.
 
@@ -1046,7 +1046,7 @@ class FunctionFieldIdeal_global(FunctionFieldIdeal_polymod):
         sage: O.ideal(y)
         Ideal (y) of Maximal order of Function field in y defined by y^2 + x^3*y + x
     """
-    def __init__(self, ring, hnf, denominator=1):
+    def __init__(self, ring, hnf, denominator=1) -> None:
         """
         Initialize.
 
@@ -1319,7 +1319,7 @@ class FunctionFieldIdealInfinite_polymod(FunctionFieldIdealInfinite):
         Ideal (1/x^4*y^2) of Maximal infinite order of Function field
         in y defined by y^3 + y^2 + 2*x^4
     """
-    def __init__(self, ring, ideal):
+    def __init__(self, ring, ideal) -> None:
         """
         Initialize this ideal.
 
@@ -1335,7 +1335,7 @@ def __init__(self, ring, ideal):
         FunctionFieldIdealInfinite.__init__(self, ring)
         self._ideal = ideal
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash of this ideal.
 
@@ -1357,7 +1357,7 @@ def __hash__(self):
         """
         return hash((self.ring(), self._ideal))
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         Return ``True`` if ``x`` is in this ideal.
 
diff --git a/src/sage/rings/function_field/ideal_rational.py b/src/sage/rings/function_field/ideal_rational.py
index 6cb21710e0d..4a99313b9a8 100644
--- a/src/sage/rings/function_field/ideal_rational.py
+++ b/src/sage/rings/function_field/ideal_rational.py
@@ -40,7 +40,7 @@ class FunctionFieldIdeal_rational(FunctionFieldIdeal):
         sage: I = O.ideal(1/(x^2+x)); I
         Ideal (1/(x^2 + x)) of Maximal order of Rational function field in x over Rational Field
     """
-    def __init__(self, ring, gen):
+    def __init__(self, ring, gen) -> None:
         """
         Initialize.
 
@@ -54,7 +54,7 @@ def __init__(self, ring, gen):
         FunctionFieldIdeal.__init__(self, ring)
         self._gen = gen
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash computed from the data.
 
@@ -65,9 +65,9 @@ def __hash__(self):
             sage: I = O.ideal(1/(x^2+x))
             sage: d = { I: 1, I^2: 2 }
         """
-        return hash( (self._ring, self._gen) )
+        return hash((self._ring, self._gen))
 
-    def __contains__(self, element):
+    def __contains__(self, element) -> bool:
         """
         Test if ``element`` is in this ideal.
 
@@ -304,7 +304,7 @@ def valuation(self, ideal):
 
         O = self.ring()
         d = ideal.denominator()
-        return self._valuation(d*ideal) - self._valuation(O.ideal(d))
+        return self._valuation(d * ideal) - self._valuation(O.ideal(d))
 
     def _valuation(self, ideal):
         """
@@ -369,7 +369,7 @@ class FunctionFieldIdealInfinite_rational(FunctionFieldIdealInfinite):
         sage: Oinf.ideal(x)
         Ideal (x) of Maximal infinite order of Rational function field in x over Finite Field of size 2
     """
-    def __init__(self, ring, gen):
+    def __init__(self, ring, gen) -> None:
         """
         Initialize.
 
@@ -384,7 +384,7 @@ def __init__(self, ring, gen):
         FunctionFieldIdealInfinite.__init__(self, ring)
         self._gen = gen
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash of this fractional ideal.
 
@@ -397,9 +397,9 @@ def __hash__(self):
             sage: J = Oinf.ideal(1/x)
             sage: d = { I: 1, J: 2 }
         """
-        return hash( (self.ring(), self._gen) )
+        return hash((self.ring(), self._gen))
 
-    def __contains__(self, element):
+    def __contains__(self, element) -> bool:
         """
         Test if ``element`` is in this ideal.
 
@@ -531,7 +531,7 @@ def is_prime(self):
             True
         """
         x = self._ring.fraction_field().gen()
-        return self._gen == 1/x
+        return self._gen == 1 / x
 
     def gen(self):
         """
diff --git a/src/sage/rings/function_field/jacobian_base.py b/src/sage/rings/function_field/jacobian_base.py
index 7274be0aea7..e4b86bf79d4 100644
--- a/src/sage/rings/function_field/jacobian_base.py
+++ b/src/sage/rings/function_field/jacobian_base.py
@@ -195,7 +195,7 @@ class JacobianGroupFunctor(ConstructionFunctor):
     """
     rank = 20
 
-    def __init__(self, base_field, field):
+    def __init__(self, base_field, field) -> None:
         """
         Initialize.
 
@@ -287,7 +287,7 @@ class JacobianGroup_base(Parent):
     """
     _embedding_map_class = None
 
-    def __init__(self, parent, function_field, base_div):
+    def __init__(self, parent, function_field, base_div) -> None:
         """
         Initialize.
 
@@ -306,7 +306,7 @@ def __init__(self, parent, function_field, base_div):
         self._genus = parent._function_field.genus()  # equals function_field.genus()
         self._base_div = base_div
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of ``self``.
 
@@ -462,8 +462,8 @@ def _bound_on_order(self):
         q = F.constant_base_field().order()
         g = self._genus
 
-        c = 2*g/(q.sqrt() - 1)
-        return integer_floor(math.exp(c)*q**g)
+        c = 2 * g / (q.sqrt() - 1)
+        return integer_floor(math.exp(c) * q**g)
 
     def order(self, algorithm='numeric'):
         """
@@ -492,50 +492,50 @@ def order(self, algorithm='numeric'):
         if algorithm == 'numeric':
             # numeric method - fast but might be inaccurate by numerical noise
             from sage.rings.qqbar import AlgebraicField
-            h = Integer(math.prod([(1-a**(-b))**m for a, m in f.change_ring(AlgebraicField()).roots()]))
+            h = Integer(math.prod([(1 - a**(-b))**m for a, m in f.change_ring(AlgebraicField()).roots()]))
             return h
 
         # algebraic method - slow
 
         es = []
         s = -1
-        for i in range(1, 2*g + 1):
-            es.append(s*f[i])
+        for i in range(1, 2 * g + 1):
+            es.append(s * f[i])
             s = -s
         es
 
         ps = [es[0]]
-        for i in range(1, 2*g):
+        for i in range(1, 2 * g):
             p = 0
             s = 1
             for j in range(i):
-                p = p + s*es[j]*ps[-j-1]
+                p = p + s * es[j] * ps[-j - 1]
                 s = -s
-            ps.append(p + s*(i + 1)*es[i])
+            ps.append(p + s * (i + 1) * es[i])
 
-        while len(ps) < b*2*g:
+        while len(ps) < b * 2 * g:
             p = 0
             s = 1
-            for j in range(2*g):
-                p = p + s*es[j]*ps[-j-1]
+            for j in range(2 * g):
+                p = p + s * es[j] * ps[-j - 1]
                 s = -s
             ps.append(p)
 
-        qs = [ps[b*(i + 1) - 1] for i in range(2*g)]
+        qs = [ps[b * (i + 1) - 1] for i in range(2 * g)]
 
         fs = [qs[0]]
-        for i in range(1, 2*g):
+        for i in range(1, 2 * g):
             k = qs[i]
             s = -1
             for j in range(i):
-                k = k + s*fs[j]*qs[i - j - 1]
+                k = k + s * fs[j] * qs[i - j - 1]
                 s = -s
-            fs.append(-s*k // (i + 1))
+            fs.append(-s * k // (i + 1))
 
         bs = [1]
         s = -1
-        for i in range(2*g):
-            bs.append(s*fs[i])
+        for i in range(2 * g):
+            bs.append(s * fs[i])
             s = -s
 
         return sum(bs)
@@ -584,7 +584,7 @@ class Jacobian_base(Parent):
         sage: F.jacobian()
         Jacobian of Function field in y defined by y^2 + y + (x^2 + 1)/x (Hess model)
     """
-    def __init__(self, function_field, base_div, **kwds):
+    def __init__(self, function_field, base_div, **kwds) -> None:
         """
         Initialize.
 
@@ -604,7 +604,7 @@ def __init__(self, function_field, base_div, **kwds):
                          base=function_field.constant_base_field(),
                          facade=True)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of ``self``.
 
@@ -682,7 +682,7 @@ def __call__(self, x):
                 K = G._function_field
                 return G.point(K.divisor_group()(x.divisor()))
         if x in F.place_set():
-            return self(x - x.degree()*self._base_place)
+            return self(x - x.degree() * self._base_place)
         if x == 0:
             return self.group().zero()
         if x in F.divisor_group():
@@ -789,7 +789,7 @@ def group(self, k_ext=None):
 
         return grp
 
-    def set_base_place(self, place):
+    def set_base_place(self, place) -> None:
         """
         Set ``place`` as the base place.
 
diff --git a/src/sage/rings/function_field/jacobian_hess.py b/src/sage/rings/function_field/jacobian_hess.py
index 8b3e0fea220..dfc3c33107f 100644
--- a/src/sage/rings/function_field/jacobian_hess.py
+++ b/src/sage/rings/function_field/jacobian_hess.py
@@ -118,7 +118,7 @@ class JacobianPoint(JacobianPoint_base):
         sage: -(dS.divisor() + ds.divisor()) == pl
         True
     """
-    def __init__(self, parent, dS, ds):
+    def __init__(self, parent, dS, ds) -> None:
         """
         Initialize.
 
@@ -135,7 +135,7 @@ def __init__(self, parent, dS, ds):
         super().__init__(parent)
         self._data = (dS, ds)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of ``self``.
 
@@ -152,7 +152,7 @@ def _repr_(self):
         divisor = (~dS).divisor() + (~ds).divisor()
         return f'[{divisor}]'
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash of ``self``.
 
@@ -445,7 +445,7 @@ class JacobianGroupEmbedding(Map):
           To:   Group of rational points of Jacobian
            over Finite Field in z3 of size 17^3 (Hess model)
     """
-    def __init__(self, base_group, extension_group):
+    def __init__(self, base_group, extension_group) -> None:
         """
         Initialize.
 
@@ -484,7 +484,7 @@ def __init__(self, base_group, extension_group):
 
         Map.__init__(self, Hom(base_group, extension_group, CommutativeAdditiveGroups()))
 
-    def _repr_type(self):
+    def _repr_type(self) -> str:
         """
         Return string representation of ``self``.
 
@@ -559,7 +559,7 @@ class JacobianGroup(UniqueRepresentation, JacobianGroup_base):
     Element = JacobianPoint
     _embedding_map_class = JacobianGroupEmbedding
 
-    def __init__(self, parent, function_field, base_div):
+    def __init__(self, parent, function_field, base_div) -> None:
         """
         Initialize.
 
@@ -585,7 +585,7 @@ def __init__(self, parent, function_field, base_div):
 
         self._base_place = None
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of ``self``.
 
@@ -865,7 +865,7 @@ class JacobianGroup_finite_field(JacobianGroup, JacobianGroup_finite_field_base)
     """
     Element = JacobianPoint_finite_field
 
-    def __init__(self, parent, function_field, base_div):
+    def __init__(self, parent, function_field, base_div) -> None:
         """
         Initialize.
 
@@ -1012,7 +1012,7 @@ class Jacobian(Jacobian_base, UniqueRepresentation):
         Jacobian of Projective Plane Curve over Finite Field of size 17
          defined by x^3 - y^2*z + 5*z^3 (Hess model)
     """
-    def __init__(self, function_field, base_div, **kwds):
+    def __init__(self, function_field, base_div, **kwds) -> None:
         """
         Initialize.
 
@@ -1032,7 +1032,7 @@ def __init__(self, function_field, base_div, **kwds):
         else:
             self._group_class = JacobianGroup
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of ``self``.
 
diff --git a/src/sage/rings/function_field/jacobian_khuri_makdisi.py b/src/sage/rings/function_field/jacobian_khuri_makdisi.py
index a1d53aff8b6..2fb4d3bed96 100644
--- a/src/sage/rings/function_field/jacobian_khuri_makdisi.py
+++ b/src/sage/rings/function_field/jacobian_khuri_makdisi.py
@@ -170,7 +170,7 @@ class JacobianPoint(JacobianPoint_base):
         [0 0 0 0 0 1 0 0 5]
         [0 0 0 0 0 0 1 0 4]
     """
-    def __init__(self, parent, w):
+    def __init__(self, parent, w) -> None:
         """
         Initialize.
 
@@ -190,7 +190,7 @@ def __init__(self, parent, w):
         w.set_immutable()
         self._w = w
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of ``self``.
 
@@ -212,7 +212,7 @@ def _repr_(self):
         """
         return f'Point of Jacobian determined by \n{self._w}'
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash of ``self``.
 
@@ -508,7 +508,7 @@ class JacobianGroupEmbedding(Map):
           To:   Group of rational points of Jacobian
                 over Finite Field in z2 of size 5^2 (Khuri-Makdisi large model)
     """
-    def __init__(self, base_group, extension_group):
+    def __init__(self, base_group, extension_group) -> None:
         """
         Initialize.
 
@@ -532,7 +532,7 @@ def __init__(self, base_group, extension_group):
 
         Map.__init__(self, Hom(base_group, extension_group, CommutativeAdditiveGroups()))
 
-    def _repr_type(self):
+    def _repr_type(self) -> str:
         """
         Return string representation of ``self``.
 
@@ -603,7 +603,7 @@ class JacobianGroup(UniqueRepresentation, JacobianGroup_base):
     Element = JacobianPoint
     _embedding_map_class = JacobianGroupEmbedding
 
-    def __init__(self, parent, function_field, base_div):
+    def __init__(self, parent, function_field, base_div) -> None:
         """
         Initialize.
 
@@ -663,7 +663,7 @@ def mu(n, m, i, j):
 
         self._base_place = None
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of ``self``.
 
@@ -855,7 +855,7 @@ class JacobianGroup_finite_field(JacobianGroup, JacobianGroup_finite_field_base)
     """
     Element = JacobianPoint_finite_field
 
-    def __init__(self, parent, function_field, base_div):
+    def __init__(self, parent, function_field, base_div) -> None:
         """
         Initialize.
 
@@ -916,7 +916,7 @@ def __iter__(self):
                         return
                     generators.append(F._places_finite(deg))
                     deg += 1
-            multiples.append((d0 + 1)*[None])
+            multiples.append((d0 + 1) * [None])
             wn = self._wd_from_divisor(new_pl.divisor())
             dn = new_pl.degree()
             wr = zero_divisor
@@ -979,7 +979,7 @@ class Jacobian(UniqueRepresentation, Jacobian_base):
         Jacobian of Projective Plane Curve over Finite Field of size 7
          defined by x^3 - y^2*z - 2*z^3 (Khuri-Makdisi large model)
     """
-    def __init__(self, function_field, base_div, model, **kwds):
+    def __init__(self, function_field, base_div, model, **kwds) -> None:
         """
         Initialize.
 
@@ -1009,7 +1009,7 @@ def __init__(self, function_field, base_div, model, **kwds):
         else:
             self._group_class = JacobianGroup
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of ``self``.
 
diff --git a/src/sage/rings/function_field/maps.py b/src/sage/rings/function_field/maps.py
index 6b37456c64c..08a057798c0 100644
--- a/src/sage/rings/function_field/maps.py
+++ b/src/sage/rings/function_field/maps.py
@@ -165,7 +165,7 @@ def _richcmp_(self, other, op):
         return richcmp((self.domain(), self.codomain()),
                        (other.domain(), other.codomain()), op)
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         r"""
         Return a hash value of this map.
 
@@ -198,7 +198,7 @@ class MapVectorSpaceToFunctionField(FunctionFieldVectorSpaceIsomorphism):
           From: Vector space of dimension 2 over Rational function field in x over Rational Field
           To:   Function field in y defined by y^2 - x*y + 4*x^3
     """
-    def __init__(self, V, K):
+    def __init__(self, V, K) -> None:
         """
         EXAMPLES::
 
@@ -307,7 +307,7 @@ class MapFunctionFieldToVectorSpace(FunctionFieldVectorSpaceIsomorphism):
           From: Function field in y defined by y^2 - x*y + 4*x^3
           To:   Vector space of dimension 2 over Rational function field in x over Rational Field
     """
-    def __init__(self, K, V):
+    def __init__(self, K, V) -> None:
         """
         Initialize.
 
@@ -380,7 +380,7 @@ class FunctionFieldMorphism(RingHomomorphism):
         Function Field endomorphism of Rational function field in x over Rational Field
           Defn: x |--> 1/x
     """
-    def __init__(self, parent, im_gen, base_morphism):
+    def __init__(self, parent, im_gen, base_morphism) -> None:
         """
         Initialize.
 
@@ -448,7 +448,7 @@ class FunctionFieldMorphism_polymod(FunctionFieldMorphism):
         sage: f(y).charpoly('y')
         y^3 + 6*x^3 + x
     """
-    def __init__(self, parent, im_gen, base_morphism):
+    def __init__(self, parent, im_gen, base_morphism) -> None:
         """
         Initialize.
 
@@ -493,7 +493,7 @@ class FunctionFieldMorphism_rational(FunctionFieldMorphism):
     """
     Morphism from a rational function field to a function field.
     """
-    def __init__(self, parent, im_gen, base_morphism):
+    def __init__(self, parent, im_gen, base_morphism) -> None:
         """
         Initialize.
 
@@ -560,7 +560,7 @@ class FunctionFieldConversionToConstantBaseField(Map):
           From: Rational function field in x over Rational Field
           To:   Rational Field
     """
-    def __init__(self, parent):
+    def __init__(self, parent) -> None:
         """
         Initialize.
 
@@ -767,7 +767,7 @@ class FunctionFieldCompletion(Map):
         b + b*t + b*t^3 + b*t^4 + (b + 1)*t^5 + (b + 1)*t^7 + b*t^9 + b*t^11
         + b*t^12 + b*t^13 + b*t^15 + b*t^16 + (b + 1)*t^17 + (b + 1)*t^19 + O(t^20)
     """
-    def __init__(self, field, place, name=None, prec=None, gen_name=None):
+    def __init__(self, field, place, name=None, prec=None, gen_name=None) -> None:
         """
         Initialize.
 
diff --git a/src/sage/rings/function_field/order.py b/src/sage/rings/function_field/order.py
index 3795e0ea25a..aa047af1667 100644
--- a/src/sage/rings/function_field/order.py
+++ b/src/sage/rings/function_field/order.py
@@ -132,7 +132,7 @@ class FunctionFieldOrder_base(CachedRepresentation, Parent):
         sage: F.maximal_order()
         Maximal order of Rational function field in y over Rational Field
     """
-    def __init__(self, field, ideal_class=FunctionFieldIdeal, category=None):
+    def __init__(self, field, ideal_class=FunctionFieldIdeal, category=None) -> None:
         """
         Initialize.
 
@@ -148,7 +148,7 @@ def __init__(self, field, ideal_class=FunctionFieldIdeal, category=None):
         self._ideal_class_ = ideal_class  # element class for parent ideal monoid
         self._field = field
 
-    def is_field(self, proof=True):
+    def is_field(self, proof: bool = True) -> bool:
         """
         Return ``False`` since orders are never fields.
 
@@ -159,7 +159,7 @@ def is_field(self, proof=True):
         """
         return False
 
-    def is_noetherian(self):
+    def is_noetherian(self) -> bool:
         """
         Return ``True`` since orders in function fields are Noetherian.
 
@@ -183,7 +183,7 @@ def function_field(self):
 
     fraction_field = function_field
 
-    def is_subring(self, other):
+    def is_subring(self, other) -> bool:
         """
         Return ``True`` if the order is a subring of the other order.
 
@@ -219,7 +219,7 @@ class FunctionFieldOrder(FunctionFieldOrder_base):
     """
     Base class for orders in function fields.
     """
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation.
 
@@ -235,7 +235,7 @@ class FunctionFieldOrderInfinite(FunctionFieldOrder_base):
     """
     Base class for infinite orders in function fields.
     """
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         EXAMPLES::
 
@@ -249,7 +249,7 @@ class FunctionFieldMaximalOrder(UniqueRepresentation, FunctionFieldOrder):
     """
     Base class of maximal orders of function fields.
     """
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of the order.
 
@@ -265,7 +265,7 @@ class FunctionFieldMaximalOrderInfinite(FunctionFieldMaximalOrder, FunctionField
     """
     Base class of maximal infinite orders of function fields.
     """
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         EXAMPLES::
 
diff --git a/src/sage/rings/function_field/order_basis.py b/src/sage/rings/function_field/order_basis.py
index 1cf89e3f966..2cd72089ab7 100644
--- a/src/sage/rings/function_field/order_basis.py
+++ b/src/sage/rings/function_field/order_basis.py
@@ -67,7 +67,7 @@ class FunctionFieldOrder_basis(FunctionFieldOrder):
         ...
         ValueError: basis (y, y, y^3, y^4, 2*x*y + (x^4 + 1)/x) is not linearly independent
     """
-    def __init__(self, basis, check=True):
+    def __init__(self, basis, check: bool = True) -> None:
         """
         Initialize.
 
@@ -104,7 +104,7 @@ def __init__(self, basis, check=True):
                 raise ValueError("basis {} is not linearly independent".format(basis))
             if to_V(field(1)) not in self._module:
                 raise ValueError("the identity element must be in the module spanned by basis {}".format(basis))
-            if not all(to_V(a*b) in self._module for a in basis for b in basis):
+            if not all(to_V(a * b) in self._module for a in basis for b in basis):
                 raise ValueError("the module generated by basis {} must be closed under multiplication".format(basis))
 
     def _element_constructor_(self, f):
@@ -240,7 +240,7 @@ def ideal(self, *gens):
                     gens = [gens]
         K = self.function_field()
 
-        return self.ideal_with_gens_over_base([b*K(g) for b in self.basis() for g in gens])
+        return self.ideal_with_gens_over_base([b * K(g) for b in self.basis() for g in gens])
 
     def polynomial(self):
         """
@@ -361,7 +361,7 @@ class FunctionFieldOrderInfinite_basis(FunctionFieldOrderInfinite):
         sage: O.basis()
         (1/x*y + 1, 1/x*y, 1/x^2*y^2, 1/x^3*y^3)
     """
-    def __init__(self, basis, check=True):
+    def __init__(self, basis, check: bool = True) -> None:
         """
         Initialize.
 
@@ -401,7 +401,7 @@ def __init__(self, basis, check=True):
                 raise ValueError("basis {} is not linearly independent".format(basis))
             if W.coordinate_vector(to(field(1))) not in self._module:
                 raise ValueError("the identity element must be in the module spanned by basis {}".format(basis))
-            if not all(W.coordinate_vector(to(a*b)) in self._module for a in basis for b in basis):
+            if not all(W.coordinate_vector(to(a * b)) in self._module for a in basis for b in basis):
                 raise ValueError("the module generated by basis {} must be closed under multiplication".format(basis))
 
     def _element_constructor_(self, f):
@@ -494,7 +494,7 @@ def ideal_with_gens_over_base(self, gens):
         gens = [F(a) for a in gens]
 
         V, from_V, to_V = F.vector_space()
-        M = V.span([to_V(b) for b in gens], base_ring=S) # not work
+        M = V.span([to_V(b) for b in gens], base_ring=S)  # not work
 
         return self.ideal_monoid().element_class(self, M)
 
@@ -533,7 +533,7 @@ def ideal(self, *gens):
                     gens = [gens]
         K = self.function_field()
 
-        return self.ideal_with_gens_over_base([b*K(g) for b in self.basis() for g in gens])
+        return self.ideal_with_gens_over_base([b * K(g) for b in self.basis() for g in gens])
 
     def polynomial(self):
         """
diff --git a/src/sage/rings/function_field/order_polymod.py b/src/sage/rings/function_field/order_polymod.py
index 2f4224f84f3..51a6251a8e3 100644
--- a/src/sage/rings/function_field/order_polymod.py
+++ b/src/sage/rings/function_field/order_polymod.py
@@ -237,7 +237,7 @@ def _ideal_from_vectors(self, vecs):
         vecs = [[(d * c).numerator() for c in v] for v in vecs]
         return self._ideal_from_vectors_and_denominator(vecs, d, check=False)
 
-    def _ideal_from_vectors_and_denominator(self, vecs, d=1, check=True):
+    def _ideal_from_vectors_and_denominator(self, vecs, d=1, check: bool = True):
         """
         Return an ideal generated as a module by vectors divided by ``d`` over
         the polynomial ring underlying the rational function field.
diff --git a/src/sage/rings/function_field/place.py b/src/sage/rings/function_field/place.py
index a17c97b7316..a4fb7c1b0e9 100644
--- a/src/sage/rings/function_field/place.py
+++ b/src/sage/rings/function_field/place.py
@@ -81,7 +81,7 @@ class FunctionFieldPlace(Element):
         sage: L.places_finite()[0]                                                      # needs sage.rings.function_field
         Place (x, y)
     """
-    def __init__(self, parent, prime):
+    def __init__(self, parent, prime) -> None:
         """
         Initialize the place.
 
@@ -96,7 +96,7 @@ def __init__(self, parent, prime):
 
         self._prime = prime
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash of the place.
 
@@ -110,7 +110,7 @@ def __hash__(self):
         """
         return hash((self.function_field(), self._prime))
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of the place.
 
@@ -129,7 +129,7 @@ def _repr_(self):
         gens_str = ', '.join(repr(g) for g in gens)
         return "Place ({})".format(gens_str)
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return the LaTeX representation of the place.
 
@@ -335,7 +335,7 @@ class PlaceSet(UniqueRepresentation, Parent):
     """
     Element = FunctionFieldPlace
 
-    def __init__(self, field):
+    def __init__(self, field) -> None:
         """
         Initialize the set of places of the function ``field``.
 
@@ -351,7 +351,7 @@ def __init__(self, field):
 
         self._field = field
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of the place.
 
diff --git a/src/sage/rings/function_field/place_polymod.py b/src/sage/rings/function_field/place_polymod.py
index c651ea2f5b3..a03ff377393 100644
--- a/src/sage/rings/function_field/place_polymod.py
+++ b/src/sage/rings/function_field/place_polymod.py
@@ -142,7 +142,7 @@ def gaps(self):
             [[1, 2, 4], [1, 2, 4], [1, 2, 4]]
         """
         if self.degree() == 1:
-            return self._gaps_rational() # faster for rational places
+            return self._gaps_rational()  # faster for rational places
         else:
             return self._gaps_wronskian()
 
@@ -182,7 +182,7 @@ def _gaps_rational(self):
         # Hess' Riemann-Roch basis algorithm stripped down for gaps computation
         def dim_RR(M):
             den = lcm([e.denominator() for e in M.list()])
-            mat = matrix(R, M.nrows(), [(den*e).numerator() for e in M.list()])
+            mat = matrix(R, M.nrows(), [(den * e).numerator() for e in M.list()])
 
             # initialise pivot_row and conflicts list
             pivot_row = [[] for i in range(n)]
@@ -191,13 +191,13 @@ def dim_RR(M):
                 bestp = -1
                 best = -1
                 for c in range(n):
-                    d = mat[i,c].degree()
+                    d = mat[i, c].degree()
                     if d >= best:
                         bestp = c
                         best = d
 
                 if best >= 0:
-                    pivot_row[bestp].append((i,best))
+                    pivot_row[bestp].append((i, best))
                     if len(pivot_row[bestp]) > 1:
                         conflicts.append(bestp)
 
@@ -205,42 +205,42 @@ def dim_RR(M):
             while conflicts:
                 c = conflicts.pop()
                 row = pivot_row[c]
-                i,ideg = row.pop()
-                j,jdeg = row.pop()
+                i, ideg = row.pop()
+                j, jdeg = row.pop()
 
                 if jdeg > ideg:
-                    i,j = j,i
-                    ideg,jdeg = jdeg,ideg
+                    i, j = j, i
+                    ideg, jdeg = jdeg, ideg
 
-                coeff = - mat[i,c].lc() / mat[j,c].lc()
+                coeff = - mat[i, c].lc() / mat[j, c].lc()
                 s = coeff * one.shift(ideg - jdeg)
 
                 mat.add_multiple_of_row(i, j, s)
 
-                row.append((j,jdeg))
+                row.append((j, jdeg))
 
                 bestp = -1
                 best = -1
                 for c in range(n):
-                    d = mat[i,c].degree()
+                    d = mat[i, c].degree()
                     if d >= best:
                         bestp = c
                         best = d
 
                 if best >= 0:
-                    pivot_row[bestp].append((i,best))
+                    pivot_row[bestp].append((i, best))
                     if len(pivot_row[bestp]) > 1:
                         conflicts.append(bestp)
 
             dim = 0
             for j in range(n):
-                i,ideg = pivot_row[j][0]
+                i, ideg = pivot_row[j][0]
                 k = den.degree() - ideg + 1
                 if k > 0:
                     dim += k
             return dim
 
-        V,fr,to = F.vector_space()
+        V, fr, to = F.vector_space()
 
         prime_inv = ~ self.prime_ideal()
         I = O.ideal(1)
@@ -264,7 +264,7 @@ def dim_RR(M):
                 else:
                     prev = dim
                 i += 1
-        else: # self is a finite place
+        else:  # self is a finite place
             Binv = B.inverse()
             while g:
                 I = I * prime_inv
@@ -305,7 +305,7 @@ def _gaps_wronskian(self):
         from sage.modules.free_module_element import vector
 
         F = self.function_field()
-        R,fr_R,to_R = self._residue_field()
+        R, fr_R, to_R = self._residue_field()
         der = F.higher_derivation()
 
         sep = self.local_uniformizer()
@@ -480,8 +480,8 @@ def _residue_field(self, name=None):
         Obasis = O.basis()
 
         M = prime.hnf()
-        R = M.base_ring() # univariate polynomial ring
-        n = M.nrows() # extension degree of the function field
+        R = M.base_ring()  # univariate polynomial ring
+        n = M.nrows()  # extension degree of the function field
 
         # Step 1: construct a vector space representing the residue field
         #
@@ -492,8 +492,8 @@ def _residue_field(self, name=None):
         # coefficients of the polynomials. V is the space of these vectors.
 
         k = F.constant_base_field()
-        degs = [M[i,i].degree() for i in range(n)]
-        deg = sum(degs) # degree of the place
+        degs = [M[i, i].degree() for i in range(n)]
+        deg = sum(degs)  # degree of the place
 
         # Let V = k**deg
 
@@ -540,7 +540,7 @@ def candidates():
             # over k (as there are finite number of intermediate fields).
             a = O._kummer_gen
             if a is not None:
-                K,fr_K,_ = self.place_below().residue_field()
+                K, fr_K, _ = self.place_below().residue_field()
                 b = fr_K(K.gen())
                 if isinstance(k, (NumberField, sage.rings.abc.AlgebraicField)):
                     kk = ZZ
@@ -559,9 +559,9 @@ def candidates():
             # Trial 3: exhaustive search in O using only polynomials
             # with coefficients 0 or 1
             for d in range(deg):
-                G = itertools.product(itertools.product([0,1],repeat=d+1), repeat=n)
+                G = itertools.product(itertools.product([0, 1], repeat=d + 1), repeat=n)
                 for g in G:
-                    gen = sum([R(c1)*c2 for c1,c2 in zip(g, Obasis)])
+                    gen = sum([R(c1) * c2 for c1, c2 in zip(g, Obasis)])
                     yield gen
 
             # Trial 4: exhaustive search in O using all polynomials
@@ -577,7 +577,7 @@ def candidates():
                     if g[j].leading_coefficient() != 1:
                         continue
 
-                    gen = sum([c1*c2 for c1,c2 in zip(g, Obasis)])
+                    gen = sum([c1 * c2 for c1, c2 in zip(g, Obasis)])
                     yield gen
 
         # Search for a primitive element. It is such an element g of O
@@ -616,7 +616,7 @@ def to_W(e):
                 prim = min_poly.roots(K)[0][0]
 
                 W, from_W, to_W = K.vector_space(k, basis=[prim**i for i in range(deg)], map=True)
-        else: # deg == 1
+        else:  # deg == 1
             K = k
 
             def from_W(e):
@@ -638,7 +638,7 @@ def from_K(e):
         # at this place and no other poles at finite places.
         p = prime.prime_below().gen().numerator()
         beta = prime._beta
-        alpha = ~p * sum(c1*c2 for c1,c2 in zip(beta, Obasis))
+        alpha = ~p * sum(c1 * c2 for c1, c2 in zip(beta, Obasis))
         alpha_powered_by_ramification_index = alpha ** prime._ramification_index
 
         def to_K(f):
@@ -648,8 +648,8 @@ def to_K(f):
 
                 # s powered by the valuation of den at the prime
                 alpha_power = alpha_powered_by_ramification_index ** den.valuation(p)
-                rn = num * alpha_power # in O
-                rd = den * alpha_power # in O but not in prime
+                rn = num * alpha_power  # in O
+                rd = den * alpha_power  # in O but not in prime
 
                 # Note that rn is not in O if and only if f is
                 # not in the valuation ring. Hence f is in the
diff --git a/src/sage/rings/function_field/place_rational.py b/src/sage/rings/function_field/place_rational.py
index 31ffd6a40b0..af479cffda4 100644
--- a/src/sage/rings/function_field/place_rational.py
+++ b/src/sage/rings/function_field/place_rational.py
@@ -142,7 +142,7 @@ def to_K(f):
             K, from_K, _to_K = O._residue_field(prime, name=name)
 
             def to_K(f):
-                if f in O: # f.denominator() is 1
+                if f in O:  # f.denominator() is 1
                     return _to_K(f.numerator())
                 else:
                     d = F(f.denominator())
@@ -157,8 +157,8 @@ def to_K(f):
                         raise TypeError("not in the valuation ring")
 
                     s = ~prime.gen()
-                    rd = d * s**dv # in O but not in prime
-                    rn = n * s**nv # in O but not in prime
+                    rd = d * s**dv  # in O but not in prime
+                    rn = n * s**nv  # in O but not in prime
                     return to_K(rn) / to_K(rd)
 
         return K, from_K, to_K
diff --git a/src/sage/rings/function_field/valuation.py b/src/sage/rings/function_field/valuation.py
index e92600e55b6..fc585d4a252 100644
--- a/src/sage/rings/function_field/valuation.py
+++ b/src/sage/rings/function_field/valuation.py
@@ -616,7 +616,7 @@ def element_with_valuation(self, s):
             return super().element_with_valuation(s)
 
         a, b = self.value_group()._element_with_valuation(constant_valuation.value_group(), s)
-        ret = self.uniformizer()**a * constant_valuation.element_with_valuation(constant_valuation.value_group().gen()*b)
+        ret = self.uniformizer()**a * constant_valuation.element_with_valuation(constant_valuation.value_group().gen() * b)
 
         return self.simplify(ret, error=s)
 
@@ -634,7 +634,7 @@ class ClassicalFunctionFieldValuation_base(DiscreteFunctionFieldValuation_base):
         sage: isinstance(v, ClassicalFunctionFieldValuation_base)
         True
     """
-    def _test_classical_residue_field(self, **options):
+    def _test_classical_residue_field(self, **options) -> None:
         r"""
         Check correctness of the residue field of a discrete valuation at a
         classical point.
@@ -680,7 +680,7 @@ class InducedRationalFunctionFieldValuation_base(FunctionFieldValuation_base):
         sage: K.<x> = FunctionField(QQ)
         sage: v = K.valuation(x^2 + 1) # indirect doctest
     """
-    def __init__(self, parent, base_valuation):
+    def __init__(self, parent, base_valuation) -> None:
         r"""
         TESTS::
 
@@ -779,7 +779,7 @@ def reduce(self, f):
         assert not ret.is_zero()
         return self.residue_field()(ret)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a printable representation of this valuation.
 
@@ -874,7 +874,7 @@ def restriction(self, ring):
             return self._base_valuation.restriction(ring)
         return super().restriction(ring)
 
-    def simplify(self, f, error=None, force=False):
+    def simplify(self, f, error=None, force: bool = False):
         r"""
         Return a simplified version of ``f``.
 
@@ -924,10 +924,10 @@ def simplify(self, f, error=None, force=False):
             # This case is not implemented yet, so we just return f which is always safe.
             return f
 
-        numerator = self.domain()(self._base_valuation.simplify(numerator, error=error+v_denominator, force=force))
-        denominator = self.domain()(self._base_valuation.simplify(denominator, error=max(v_denominator, error - v_numerator + 2*v_denominator), force=force))
+        numerator = self.domain()(self._base_valuation.simplify(numerator, error=error + v_denominator, force=force))
+        denominator = self.domain()(self._base_valuation.simplify(denominator, error=max(v_denominator, error - v_numerator + 2 * v_denominator), force=force))
 
-        ret = numerator/denominator
+        ret = numerator / denominator
         assert self(ret - f) > error
         return ret
 
@@ -991,7 +991,7 @@ class FiniteRationalFunctionFieldValuation(InducedRationalFunctionFieldValuation
         sage: q = L.valuation(x^6 - t); q
         (x^6 + 2*t)-adic valuation
     """
-    def __init__(self, parent, base_valuation):
+    def __init__(self, parent, base_valuation) -> None:
         r"""
         TESTS::
 
@@ -1018,7 +1018,7 @@ class NonClassicalRationalFunctionFieldValuation(InducedRationalFunctionFieldVal
         sage: w = K.valuation(v); w  # indirect doctest
         2-adic valuation
     """
-    def __init__(self, parent, base_valuation):
+    def __init__(self, parent, base_valuation) -> None:
         r"""
         TESTS:
 
@@ -1092,7 +1092,7 @@ class FunctionFieldFromLimitValuation(FiniteExtensionFromLimitValuation, Discret
         sage: w = v.extension(L); w                                                     # needs sage.rings.function_field
         (x - 1)-adic valuation
     """
-    def __init__(self, parent, approximant, G, approximants):
+    def __init__(self, parent, approximant, G, approximants) -> None:
         r"""
         TESTS::
 
@@ -1157,7 +1157,7 @@ class FunctionFieldMappedValuation_base(FunctionFieldValuation_base, MappedValua
         sage: v = K.valuation(1/x); v
         Valuation at the infinite place
     """
-    def __init__(self, parent, base_valuation, to_base_valuation_domain, from_base_valuation_domain):
+    def __init__(self, parent, base_valuation, to_base_valuation_domain, from_base_valuation_domain) -> None:
         r"""
         TESTS::
 
@@ -1225,7 +1225,7 @@ def scale(self, scalar):
             return self.domain().valuation((self._base_valuation.scale(scalar), self._to_base, self._from_base))
         return super().scale(scalar)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a printable representation of this valuation.
 
@@ -1273,7 +1273,7 @@ class FunctionFieldMappedValuationRelative_base(FunctionFieldMappedValuation_bas
         sage: v = K.valuation(1/x); v
         Valuation at the infinite place
     """
-    def __init__(self, parent, base_valuation, to_base_valuation_domain, from_base_valuation_domain):
+    def __init__(self, parent, base_valuation, to_base_valuation_domain, from_base_valuation_domain) -> None:
         r"""
         TESTS::
 
@@ -1318,7 +1318,7 @@ class RationalFunctionFieldMappedValuation(FunctionFieldMappedValuationRelative_
         [ Gauss valuation induced by 2-adic valuation, v(x) = 1 ]
         (in Rational function field in x over Rational Field after x |--> 1/x)
     """
-    def __init__(self, parent, base_valuation, to_base_valuation_doain, from_base_valuation_domain):
+    def __init__(self, parent, base_valuation, to_base_valuation_doain, from_base_valuation_domain) -> None:
         r"""
         TESTS::
 
@@ -1344,7 +1344,7 @@ class InfiniteRationalFunctionFieldValuation(FunctionFieldMappedValuationRelativ
         sage: K.<x> = FunctionField(QQ)
         sage: v = K.valuation(1/x)  # indirect doctest
     """
-    def __init__(self, parent):
+    def __init__(self, parent) -> None:
         r"""
         TESTS::
 
@@ -1355,11 +1355,11 @@ def __init__(self, parent):
             True
         """
         x = parent.domain().gen()
-        FunctionFieldMappedValuationRelative_base.__init__(self, parent, FunctionFieldValuation(parent.domain(), x), parent.domain().hom([1/x]), parent.domain().hom([1/x]))
+        FunctionFieldMappedValuationRelative_base.__init__(self, parent, FunctionFieldValuation(parent.domain(), x), parent.domain().hom([1 / x]), parent.domain().hom([1 / x]))
         RationalFunctionFieldValuation_base.__init__(self, parent)
         ClassicalFunctionFieldValuation_base.__init__(self, parent)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a printable representation of this valuation.
 
@@ -1401,7 +1401,7 @@ class FunctionFieldExtensionMappedValuation(FunctionFieldMappedValuationRelative
         sage: isinstance(w, FunctionFieldExtensionMappedValuation)                      # needs sage.rings.function_field
         True
     """
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a printable representation of this valuation.
 
diff --git a/src/sage/rings/function_field/valuation_ring.py b/src/sage/rings/function_field/valuation_ring.py
index 9269d21ea2d..1c29214bea5 100644
--- a/src/sage/rings/function_field/valuation_ring.py
+++ b/src/sage/rings/function_field/valuation_ring.py
@@ -91,7 +91,7 @@ class FunctionFieldValuationRing(UniqueRepresentation, Parent):
         sage: p.valuation_ring()
         Valuation ring at Place (x, x*y)
     """
-    def __init__(self, field, place, category=None):
+    def __init__(self, field, place, category=None) -> None:
         """
         Initialize.
 
@@ -134,7 +134,7 @@ def _element_constructor_(self, x):
         else:
             raise TypeError
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of the valuation ring.
 
@@ -206,6 +206,6 @@ def residue_field(self, name=None):
         from .maps import FunctionFieldRingMorphism as morphism
 
         k, from_k, to_k = self._place._residue_field(name=name)
-        mor_from_k = morphism(Hom(k,self), from_k)
-        mor_to_k = morphism(Hom(self,k), to_k)
+        mor_from_k = morphism(Hom(k, self), from_k)
+        mor_to_k = morphism(Hom(self, k), to_k)
         return k, mor_from_k, mor_to_k