Merge branch 'ore_not_extension' into useless_limitations

Author: Xavier Caruso

src/sage/categories/drinfeld_modules.py

@@ -70,7 +70,7 @@ class DrinfeldModules(Category_over_base_ring):
         sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
         sage: C = phi.category()
         sage: C
-        Category of Drinfeld modules over Finite Field in z of size 11^4 over its base
+        Category of Drinfeld modules over Finite Field in z of size 11^4
 
     The output tells the user that the category is only defined by its
     base.
@@ -88,7 +88,7 @@ class DrinfeldModules(Category_over_base_ring):
         sage: C.base_morphism()
         Ring morphism:
           From: Univariate Polynomial Ring in T over Finite Field of size 11
-          To:   Finite Field in z of size 11^4 over its base
+          To:   Finite Field in z of size 11^4
           Defn: T |--> z^3 + 7*z^2 + 6*z + 10
 
     The so-called constant coefficient --- which is the same for all
@@ -123,7 +123,7 @@ class DrinfeldModules(Category_over_base_ring):
         True
 
         sage: C.ore_polring()
-        Ore Polynomial Ring in t over Finite Field in z of size 11^4 over its base twisted by Frob
+        Ore Polynomial Ring in t over Finite Field in z of size 11^4 twisted by z |--> z^11
         sage: C.ore_polring() is phi.ore_polring()
         True
 
@@ -165,20 +165,24 @@ class DrinfeldModules(Category_over_base_ring):
         sage: K.<z> = Fq.extension(4)
         sage: from sage.categories.drinfeld_modules import DrinfeldModules
         sage: base = Hom(A, K)(0)
-        sage: C = DrinfeldModules(base)
+        sage: C = DrinfeldModules(base)  # known bug (blankline)
+        <BLANKLINE>
         Traceback (most recent call last):
         ...
         TypeError: base field must be a ring extension
 
-    ::
+    Note that `C.base_morphism()` has codomain `K` while
+    the defining morphism of `C.base()` has codomain `K` viewed
+    as an `A`-field. Thus, they differ::
 
         sage: C.base().defining_morphism() == C.base_morphism()
-        True
+        False
 
     ::
 
         sage: base = Hom(A, A)(1)
-        sage: C = DrinfeldModules(base)
+        sage: C = DrinfeldModules(base)  # known bug (blankline)
+        <BLANKLINE>
         Traceback (most recent call last):
         ...
         TypeError: base field must be a ring extension
@@ -203,7 +207,7 @@ class DrinfeldModules(Category_over_base_ring):
         TypeError: function ring base must be a finite field
     """
 
-    def __init__(self, base_field, name='t'):
+    def __init__(self, base_morphism, name='t'):
         r"""
         Initialize ``self``.
 
@@ -223,34 +227,23 @@ def __init__(self, base_field, name='t'):
             sage: p_root = z^3 + 7*z^2 + 6*z + 10
             sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
             sage: C = phi.category()
-            sage: ore_polring.<t> = OrePolynomialRing(phi.base(), phi.base().frobenius_endomorphism())
+            sage: ore_polring.<t> = OrePolynomialRing(K, K.frobenius_endomorphism())
             sage: C._ore_polring is ore_polring
             True
-            sage: i = phi.base().coerce_map_from(K)
-            sage: base_morphism = Hom(A, K)(p_root)
-            sage: C.base() == K.over(base_morphism)
-            True
-            sage: C._base_morphism == i * base_morphism
-            True
             sage: C._function_ring is A
             True
-            sage: C._constant_coefficient == base_morphism(T)
+            sage: C._constant_coefficient == C._base_morphism(T)
             True
             sage: C._characteristic(C._constant_coefficient)
             0
         """
-        # Check input is a ring extension
-        if not isinstance(base_field, RingExtension_generic):
-            raise TypeError('base field must be a ring extension')
-        base_morphism = base_field.defining_morphism()
         self._base_morphism = base_morphism
+        function_ring = self._function_ring = base_morphism.domain()
+        base_field = self._base_field = base_morphism.codomain()
         # Check input is a field
         if not base_field.is_field():
             raise TypeError('input must be a field')
-        self._base_field = base_field
-        self._function_ring = base_morphism.domain()
         # Check domain of base morphism is Fq[T]
-        function_ring = self._function_ring
         if not isinstance(function_ring, PolynomialRing_generic):
             raise NotImplementedError('function ring must be a polynomial '
                                       'ring')
@@ -262,7 +255,7 @@ def __init__(self, base_field, name='t'):
         Fq = function_ring_base
         A = function_ring
         T = A.gen()
-        K = base_field  # A ring extension
+        K = base_field
         # Build K{t}
         d = log(Fq.cardinality(), Fq.characteristic())
         tau = K.frobenius_endomorphism(d)
@@ -284,7 +277,7 @@ def __init__(self, base_field, name='t'):
         i = A.coerce_map_from(Fq)
         Fq_to_K = self._base_morphism * i
         self._base_over_constants_field = base_field.over(Fq_to_K)
-        super().__init__(base=base_field)
+        super().__init__(base=base_field.over(base_morphism))
 
     def _latex_(self):
         r"""
@@ -321,7 +314,7 @@ def _repr_(self):
             sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
             sage: C = phi.category()
             sage: C
-            Category of Drinfeld modules over Finite Field in z of size 11^4 over its base
+            Category of Drinfeld modules over Finite Field in z of size 11^4
         """
         return f'Category of Drinfeld modules over {self._base_field}'
 
@@ -378,7 +371,7 @@ def base_morphism(self):
             sage: C.base_morphism()
             Ring morphism:
               From: Univariate Polynomial Ring in T over Finite Field of size 11
-              To:   Finite Field in z of size 11^4 over its base
+              To:   Finite Field in z of size 11^4
               Defn: T |--> z^3 + 7*z^2 + 6*z + 10
 
             sage: C.constant_coefficient() == C.base_morphism()(T)
@@ -421,7 +414,7 @@ def characteristic(self):
 
         ::
 
-            sage: psi = DrinfeldModule(A, [Frac(A).gen(), 1])
+            sage: psi = DrinfeldModule(A, [T, 1])
             sage: C = psi.category()
             sage: C.characteristic()
             0
@@ -517,7 +510,7 @@ def ore_polring(self):
             sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
             sage: C = phi.category()
             sage: C.ore_polring()
-            Ore Polynomial Ring in t over Finite Field in z of size 11^4 over its base twisted by Frob
+            Ore Polynomial Ring in t over Finite Field in z of size 11^4 twisted by z |--> z^11
         """
         return self._ore_polring
 
@@ -595,7 +588,7 @@ def base(self):
 
             The base can be infinite::
 
-                sage: sigma = DrinfeldModule(A, [Frac(A).gen(), 1])
+                sage: sigma = DrinfeldModule(A, [T, 1])
                 sage: sigma.base()
                 Fraction Field of Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2 over its base
             """
@@ -615,17 +608,16 @@ def base_morphism(self):
                 sage: phi.base_morphism()
                 Ring morphism:
                   From: Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
-                  To:   Finite Field in z12 of size 5^12 over its base
+                  To:   Finite Field in z12 of size 5^12
                   Defn: T |--> 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12
 
             The base field can be infinite::
 
-                sage: sigma = DrinfeldModule(A, [Frac(A).gen(), 1])
+                sage: sigma = DrinfeldModule(A, [T, 1])
                 sage: sigma.base_morphism()
-                Ring morphism:
+                Coercion map:
                   From: Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
-                  To:   Fraction Field of Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2 over its base
-                  Defn: T |--> T
+                  To:   Fraction Field of Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
             """
             return self.category().base_morphism()
 
@@ -668,8 +660,7 @@ def characteristic(self):
             ::
 
                 sage: B.<Y> = Fq[]
-                sage: L = Frac(B)
-                sage: psi = DrinfeldModule(A, [L(1), 0, 0, L(1)])
+                sage: psi = DrinfeldModule(A, [B(1), 0, 0, 1])
                 sage: psi.characteristic()
                 Traceback (most recent call last):
                 ...
@@ -753,7 +744,7 @@ def ore_polring(self):
                 sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
                 sage: S = phi.ore_polring()
                 sage: S
-                Ore Polynomial Ring in t over Finite Field in z12 of size 5^12 over its base twisted by Frob^2
+                Ore Polynomial Ring in t over Finite Field in z12 of size 5^12 twisted by z12 |--> z12^(5^2)
 
             The Ore polynomial ring can also be retrieved from the category
             of the Drinfeld module::
@@ -782,7 +773,7 @@ def ore_variable(self):
                 sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
 
                 sage: phi.ore_polring()
-                Ore Polynomial Ring in t over Finite Field in z12 of size 5^12 over its base twisted by Frob^2
+                Ore Polynomial Ring in t over Finite Field in z12 of size 5^12 twisted by z12 |--> z12^(5^2)
                 sage: phi.ore_variable()
                 t
             """