sagemathgh-40421: Limit the use of RingExtensions in Drinfeld modules
    
Behaviour before this PR:
If `phi` is a Drinfeld module, the method `phi.ore_polring()` used to
return an Ore polynomial ring with coefficients in `RingExtension`.

This causes a lot of headache because currently an instance of
`RingExtension`  does not always belong to the appropriate category and,
consequently, does not implement all relevant methods and/or is not
recognized as what it actually is (e.g. a finite field).

In this PR, we propose a simple fix: we let `phi.ore_polring()` return
an Ore polynomial ring over the underlying regular field. This only
changes the way things are printed, but does not alter the user
interface.

We will maybe come back later to the original implementation when
`RingExtension` will be fixed (which does not seem to be
straightforward).

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.
    
URL: sagemath#40421
Reported by: Xavier Caruso
Reviewer(s): Antoine Leudière

Author: Release Manager

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}'
 
@@ -402,7 +395,7 @@ 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)
@@ -541,7 +534,7 @@ 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
 
@@ -666,17 +659,16 @@ 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.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()
 
@@ -804,7 +796,7 @@ 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::
@@ -833,7 +825,7 @@ 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
             """

src/sage/rings/function_field/drinfeld_modules/charzero_drinfeld_module.py

@@ -504,7 +504,6 @@ def coefficient_in_function_ring(self, n):
         """
         A = self.function_ring()
         g = self.coefficient(n)
-        g = g.backend(force=True)
         if g.denominator().is_one():
             return A(g.numerator().list())
         else:
@@ -550,7 +549,6 @@ def coefficients_in_function_ring(self, sparse=True):
         A = self.function_ring()
         gs = []
         for g in self.coefficients(sparse):
-            g = g.backend(force=True)
             if g.denominator().is_one():
                 gs.append(A(g.numerator().list()))
             else:

src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py

@@ -194,7 +194,7 @@ class DrinfeldModule(Parent, UniqueRepresentation):
     :class:`sage.categories.drinfeld_modules.DrinfeldModules`)::
 
         sage: phi.category()
-        Category of Drinfeld modules over Finite Field in z of size 3^12 over its base
+        Category of Drinfeld modules over Finite Field in z of size 3^12
         sage: phi.category() is psi.category()
         False
         sage: phi.category() is rho.category()
@@ -219,7 +219,7 @@ class DrinfeldModule(Parent, UniqueRepresentation):
         sage: phi.base_morphism()
         Ring morphism:
           From: Univariate Polynomial Ring in T over Finite Field in z2 of size 3^2
-          To:   Finite Field in z of size 3^12 over its base
+          To:   Finite Field in z of size 3^12
           Defn: T |--> z
 
     Note that the base field is *not* the field `K`. Rather, it is a
@@ -237,14 +237,13 @@ class DrinfeldModule(Parent, UniqueRepresentation):
         sage: phi.base_morphism()
         Ring morphism:
           From: Univariate Polynomial Ring in T over Finite Field in z2 of size 3^2
-          To:   Finite Field in z of size 3^12 over its base
+          To:   Finite Field in z of size 3^12
           Defn: T |--> z
 
     ::
 
         sage: phi.ore_polring()  # K{t}
-        Ore Polynomial Ring in t over Finite Field in z of size 3^12 over its base
-         twisted by Frob^2
+        Ore Polynomial Ring in t over Finite Field in z of size 3^12 twisted by z |--> z^(3^2)
 
     ::
 
@@ -268,9 +267,8 @@ class DrinfeldModule(Parent, UniqueRepresentation):
         sage: phi.morphism()  # The Drinfeld module as a morphism
         Ring morphism:
           From: Univariate Polynomial Ring in T over Finite Field in z2 of size 3^2
-          To:   Ore Polynomial Ring in t
-                over Finite Field in z of size 3^12 over its base
-                twisted by Frob^2
+          To:   Ore Polynomial Ring in t over Finite Field in z of size 3^12
+                twisted by z |--> z^(3^2)
           Defn: T |--> t^2 + t + z
 
     One can compute the rank and height::
@@ -578,42 +576,37 @@ def __classcall_private__(cls, function_ring, gen, name='t'):
         if isinstance(gen, OrePolynomial):
             ore_polring = gen.parent()
             # Base ring without morphism structure:
-            base_field_noext = ore_polring.base()
+            base_field = ore_polring.base()
             name = ore_polring.variable_name()
         # `gen` is a list of coefficients (function_ring = Fq[T]):
         elif isinstance(gen, (list, tuple)):
             ore_polring = None
             # Base ring without morphism structure:
-            base_field_noext = Sequence(gen).universe()
+            base_field = Sequence(gen).universe()
         else:
             raise TypeError('generator must be list of coefficients or Ore '
                             'polynomial')
         # Constant coefficient must be nonzero:
         if gen[0].is_zero():
             raise ValueError('constant coefficient must be nonzero')
         # The coefficients are in a base field that has coercion from Fq:
-        if not (hasattr(base_field_noext, 'has_coerce_map_from') and
-                base_field_noext.has_coerce_map_from(function_ring.base_ring())):
+        if not (hasattr(base_field, 'has_coerce_map_from') and
+                base_field.has_coerce_map_from(function_ring.base_ring())):
             raise ValueError('function ring base must coerce into base field')
 
         # Build the category
         T = function_ring.gen()
-        if isinstance(base_field_noext, RingExtension_generic):
-            base_field = base_field_noext
-        elif base_field_noext.has_coerce_map_from(function_ring) \
-                and T == gen[0]:
-            base_morphism = base_field_noext.coerce_map_from(function_ring)
-            base_field = base_field_noext.over(base_morphism)
+        if base_field.has_coerce_map_from(function_ring) and T == gen[0]:
+            base_morphism = base_field.coerce_map_from(function_ring)
         else:
-            base_morphism = Hom(function_ring, base_field_noext)(gen[0])
-            base_field = base_field_noext.over(base_morphism)
+            base_morphism = Hom(function_ring, base_field)(gen[0])
 
         # This test is also done in the category. We put it here also
         # to have a friendlier error message
         if not base_field.is_field():
             raise ValueError('generator coefficients must live in a field')
 
-        category = DrinfeldModules(base_field, name=name)
+        category = DrinfeldModules(base_morphism, name=name)
 
         # Check gen as Ore polynomial
         ore_polring = category.ore_polring()  # Sanity cast
@@ -622,12 +615,11 @@ def __classcall_private__(cls, function_ring, gen, name='t'):
             raise ValueError('generator must have positive degree')
 
         # Instantiate the appropriate class:
-        backend = base_field.backend(force=True)
-        if backend.is_finite():
+        if base_field.is_finite():
             from sage.rings.function_field.drinfeld_modules.finite_drinfeld_module import DrinfeldModule_finite
             return DrinfeldModule_finite(gen, category)
-        if isinstance(backend, FractionField_generic):
-            ring = backend.ring()
+        if isinstance(base_field, FractionField_generic):
+            ring = base_field.ring()
             if (isinstance(ring, PolynomialRing_generic)
             and ring.base_ring() is function_ring_base
             and base_morphism(T) == ring.gen()):
@@ -1792,7 +1784,7 @@ def morphism(self):
             Ring morphism:
               From: Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
               To:   Ore Polynomial Ring in t over Finite Field in z12 of size 5^12
-                    over its base twisted by Frob^2
+                    twisted by z12 |--> z12^(5^2)
               Defn: T |--> z12^5*t^2 + z12^3*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
             sage: from sage.rings.morphism import RingHomomorphism

src/sage/rings/function_field/drinfeld_modules/morphism.py

@@ -593,7 +593,7 @@ def _motive_matrix(self):
         # The next rows:
         # each row is obtained from the previous one by
         # applying the semi-linear transformation f |-> t*f
-        inv = K(phiT[r]).inverse()
+        inv = ~phiT[r]
         B = inv * phiT
         T = KT.gen()
         for i in range(1, r):