gh-35770: cylint: remove some unused variables in rings/
    
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### 📚 Description

remove some of the many unused variables in pyx files in the `rings`
folder, found using `cython-lint`

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URL: #35770
Reported by: Frédéric Chapoton
Reviewer(s): Frédéric Chapoton, Matthias Köppe

Author: Release Manager

src/sage/rings/bernoulli_mod_p.pyx

@@ -311,8 +311,6 @@ def bernoulli_mod_p_single(long p, long k):
     if not sage.arith.all.is_prime(p):
         raise ValueError("p (=%s) must be a prime" % p)
 
-    R = Integers(p)
-
     cdef long x = bernmm_bern_modp(p, k)
     if x == -1:
         raise ArithmeticError("B_k is not integral at p")

src/sage/rings/complex_arb.pyx

@@ -154,7 +154,7 @@ cimport sage.rings.abc
 cimport sage.rings.rational
 
 from cpython.int cimport PyInt_AS_LONG
-from cpython.object cimport Py_LT, Py_LE, Py_EQ, Py_NE, Py_GT, Py_GE
+from cpython.object cimport Py_EQ, Py_NE
 from cpython.complex cimport PyComplex_FromDoubles
 
 from sage.ext.stdsage cimport PY_NEW
@@ -1402,7 +1402,6 @@ cdef class ComplexBall(RingElement):
         """
         cdef fmpz_t tmpz
         cdef fmpq_t tmpq
-        cdef long myprec
         cdef bint cplx = False
 
         Element.__init__(self, parent)
@@ -2426,17 +2425,15 @@ cdef class ComplexBall(RingElement):
             False
         """
         cdef ComplexBall lt, rt
-        cdef acb_t difference
 
         lt = left
         rt = right
 
         if op == Py_EQ:
             return acb_eq(lt.value, rt.value)
-        elif op == Py_NE:
+        if op == Py_NE:
             return acb_ne(lt.value, rt.value)
-        elif op == Py_GT or op == Py_GE or op == Py_LT or op == Py_LE:
-            raise TypeError("No order is defined for ComplexBalls.")
+        raise TypeError("No order is defined for ComplexBalls.")
 
     def identical(self, ComplexBall other):
         """

src/sage/rings/complex_mpc.pyx

@@ -816,7 +816,6 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
         """
         # This should not be called except when the number is being created.
         # Complex Numbers are supposed to be immutable.
-        cdef RealNumber x
         cdef mpc_rnd_t rnd
         rnd =(<MPComplexField_class>self._parent).__rnd
         if y is None:
@@ -2338,12 +2337,11 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
             sage: u.agm(v, algorithm="optimal")
             -0.410522769709397 + 4.60061063922097*I
         """
-        if algorithm=="pari":
+        if algorithm == "pari":
             t = self._parent(right).__pari__()
             return self._parent(self.__pari__().agm(t))
 
         cdef MPComplexNumber a, b, d, s, res
-        cdef mpfr_t sn,dn
         cdef mp_exp_t rel_prec
         cdef bint optimal = algorithm == "optimal"
 

src/sage/rings/fast_arith.pyx

@@ -146,7 +146,6 @@ cpdef prime_range(start, stop=None, algorithm=None, bint py_ints=False):
     - Kevin Stueve (added primes iterator option) 2010-10-16
     - Robert Bradshaw (speedup using Pari prime table, py_ints option)
     """
-    cdef Integer z
     # input to pari.init_primes cannot be greater than 436273290 (hardcoded bound)
     DEF init_primes_max = 436273290
     DEF small_prime_max = 436273009  #  a prime < init_primes_max (preferably the largest)

src/sage/rings/finite_rings/element_base.pyx

@@ -81,7 +81,6 @@ cdef class FiniteRingElement(CommutativeRingElement):
             sage: r = a._nth_root_common(29*283*3539*12345, False, "Johnston", False)
             sage: r**(29*283*3539*12345) == a
             True
-
         """
         K = self.parent()
         q = K.order()
@@ -96,7 +95,7 @@ cdef class FiniteRingElement(CommutativeRingElement):
         if gcd == q-1:
             if all: return []
             else: raise ValueError("no nth root")
-        gcd, alpha, beta = n.xgcd(q-1) # gcd = alpha*n + beta*(q-1), so 1/n = alpha/gcd (mod q-1)
+        gcd, alpha, _ = n.xgcd(q-1)  # gcd = alpha*n + beta*(q-1), so 1/n = alpha/gcd (mod q-1)
         if gcd == 1:
             return [self**alpha] if all else self**alpha
 
@@ -406,12 +405,11 @@ cdef class FinitePolyExtElement(FiniteRingElement):
             sage: e._vector_(reverse=True)
             (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1)
         """
-        #vector(foo) might pass in ZZ
+        # vector(foo) might pass in ZZ
         if isinstance(reverse, Parent):
             raise TypeError("Base field is fixed to prime subfield.")
 
         k = self.parent()
-        p = self.polynomial()
         ret = self.polynomial().padded_list(k.degree())
 
         if reverse:
@@ -422,7 +420,9 @@ cdef class FinitePolyExtElement(FiniteRingElement):
         r"""
         Return the matrix of left multiplication by the element on
         the power basis `1, x, x^2, \ldots, x^{d-1}` for the field
-        extension.  Thus the \emph{columns} of this matrix give the images
+        extension.
+
+        Thus the \emph{columns} of this matrix give the images
         of each of the `x^i`.
 
         INPUT:

src/sage/rings/finite_rings/integer_mod.pyx

@@ -147,7 +147,6 @@ def Mod(n, m, parent=None):
         return n
 
     # m is non-zero, so return n mod m
-    cdef IntegerMod_abstract x
     if parent is None:
         from .integer_mod_ring import IntegerModRing
         parent = IntegerModRing(m)
@@ -1524,7 +1523,6 @@ cdef class IntegerMod_abstract(FiniteRingElement):
                     return K(p**(pval // n) * mod(upart, p**(k-pval)).nth_root(n, algorithm=algorithm).lift())
             from sage.rings.padics.factory import ZpFM
             R = ZpFM(p,k)
-            self_orig = self
             if p == 2:
                 sign = [1]
                 if self % 4 == 3:
@@ -4034,9 +4032,6 @@ cpdef square_root_mod_prime(IntegerMod_abstract a, p=None):
     cdef double bits = log2(float(p))
     cdef long r, m
 
-    cdef Integer resZ
-
-
     if p_mod_16 % 2 == 0:  # p == 2
         return a
 

src/sage/rings/function_field/element.pyx

@@ -655,15 +655,15 @@ cdef class FunctionFieldElement(FieldElement):
             sage: (y/x + 1).evaluate(p)                                                 # optional - sage.rings.finite_rings sage.rings.function_field
             1
         """
-        R, fr_R, to_R = place._residue_field()
+        R, _, to_R = place._residue_field()
 
         v = self.valuation(place)
         if v > 0:
             return R.zero()
-        elif v  == 0:
+        if v  == 0:
             return to_R(self)
-        else: # v < 0
-            raise ValueError('has a pole at the place')
+        # v < 0
+        raise ValueError('has a pole at the place')
 
     cpdef bint is_nth_power(self, n):
         r"""

src/sage/rings/function_field/element_polymod.pyx

@@ -298,7 +298,7 @@ cdef class FunctionFieldElement_polymod(FunctionFieldElement):
         # reduce to the separable case
         poly = self._parent._polynomial
         if not poly.gcd(poly.derivative()).is_one():
-            L, from_L, to_L = self._parent.separable_model(('t', 'w'))
+            _, from_L, to_L = self._parent.separable_model(('t', 'w'))
             return from_L(to_L(self).nth_root(n))
 
         constant_base_field = self._parent.constant_base_field()
@@ -347,7 +347,7 @@ cdef class FunctionFieldElement_polymod(FunctionFieldElement):
         # reduce to the separable case
         poly = self._parent._polynomial
         if not poly.gcd(poly.derivative()).is_one():
-            L, from_L, to_L = self._parent.separable_model(('t', 'w'))
+            _, _, to_L = self._parent.separable_model(('t', 'w'))
             return to_L(self).is_nth_power(n)
 
         constant_base_field = self._parent.constant_base_field()

src/sage/rings/integer_ring.pyx

@@ -413,7 +413,7 @@ cdef class IntegerRing_class(PrincipalIdealDomain):
             return self
 
         if isinstance(x, NumberFieldElement_base):
-            K, from_K = parent(x).subfield(x)
+            K, _ = parent(x).subfield(x)
             return K.order(K.gen())
 
         return PrincipalIdealDomain.__getitem__(self, x)
@@ -1651,7 +1651,7 @@ def crt_basis(X, xgcd=None):
     for i in range(len(X)):
         p = X[i]
         others = P // p
-        g, s, t = p.xgcd(others)
+        g, _, t = p.xgcd(others)
         if g != ONE:
             raise ArithmeticError("the elements of the list X must be coprime in pairs")
         Y.append(t * others)

src/sage/rings/laurent_series_ring_element.pyx

@@ -1255,7 +1255,6 @@ cdef class LaurentSeries(AlgebraElement):
             deg = prec - 1
 
         cdef long i
-        cdef int c
         for i in range(val, deg + 1):
             li = self[i]
             ri = right[i]