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Distinguish between base and A_field for Drinfeld modules #41289

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c492040
distinguish between base and A_field
Dec 13, 2025
abec418
Merge branch 'develop' into drinfeld_base
Dec 17, 2025
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35 changes: 11 additions & 24 deletions src/sage/categories/drinfeld_modules.py
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Original file line number Diff line number Diff line change
Expand Up @@ -79,7 +79,7 @@ class DrinfeldModules(Category_over_base_ring):
The base field is retrieved using the method :meth:`base`::

sage: C.base()
Finite Field in z of size 11^4 over its base
Finite Field in z of size 11^4

Equivalently, one can use :meth:`base_morphism` to retrieve the base
morphism::
Expand Down Expand Up @@ -170,13 +170,6 @@ class DrinfeldModules(Category_over_base_ring):
...
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()
False

::

sage: base = Hom(A, A)(1)
Expand Down Expand Up @@ -276,7 +269,7 @@ def __init__(self, base_morphism, name='τ'):
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.over(base_morphism))
super().__init__(base=base_field)

def _latex_(self):
r"""
Expand Down Expand Up @@ -377,7 +370,7 @@ def A_field(self):
sage: C.A_field()
Finite Field in z of size 5^12 over its base
"""
return self.base()
return self.base().over(self._base_morphism)

def base_morphism(self):
r"""
Expand Down Expand Up @@ -461,7 +454,7 @@ def constant_coefficient(self):
sage: C = phi.category()
sage: C.constant_coefficient()
z^3 + 7*z^2 + 6*z + 10
sage: C.constant_coefficient() == C.base()(T)
sage: C.constant_coefficient() == C.A_field()(T)
True
"""
return self._constant_coefficient
Expand Down Expand Up @@ -616,15 +609,8 @@ def A_field(self):

def base(self):
r"""
Return the underlying `A`-field of this Drinfeld module,
viewed as an algebra over the function ring `A`.

This is an instance of the class
:class:`sage.rings.ring_extension.RingExtension`.

.. NOTE::

This method has the same behavior as :meth:`A_field`.
Return the field over which this Drinfeld module
is defined.

EXAMPLES::

Expand All @@ -634,14 +620,15 @@ def base(self):
sage: p_root = 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: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
sage: phi.base()
Finite Field in z12 of size 5^12 over its base
Finite Field in z12 of size 5^12

The base can be infinite::

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
Fraction Field of Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
"""
# should we add a deprecation?
return self.category().base()

def base_morphism(self):
Expand Down Expand Up @@ -758,8 +745,8 @@ def constant_coefficient(self):
`\gamma(T)`::

sage: C = phi.category()
sage: base = C.base()
sage: base(T) == phi.constant_coefficient()
sage: F = C.A_field()
sage: F(T) == phi.constant_coefficient()
True

Naturally, two Drinfeld modules in the same category have the
Expand Down
4 changes: 2 additions & 2 deletions src/sage/rings/function_field/drinfeld_modules/action.py
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Expand Up @@ -59,7 +59,7 @@ class DrinfeldModuleAction(Action):
sage: phi = DrinfeldModule(A, [z, 0, 0, 1])
sage: action = phi.action()
sage: action
Action on Finite Field in z of size 11^2 over its base
Action on Finite Field in z of size 11^2
induced by Drinfeld module defined by T |--> τ^3 + z

The action on elements is computed as follows::
Expand Down Expand Up @@ -174,7 +174,7 @@ def _repr_(self) -> str:
sage: phi = DrinfeldModule(A, [z, 0, 0, 1])
sage: action = phi.action()
sage: action
Action on Finite Field in z of size 11^2 over its base induced by Drinfeld module defined by T |--> τ^3 + z
Action on Finite Field in z of size 11^2 induced by Drinfeld module defined by T |--> τ^3 + z
"""
return f'Action on {self._base} induced by ' \
f'{self._drinfeld_module}'
Expand Down
19 changes: 5 additions & 14 deletions src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
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Expand Up @@ -101,7 +101,7 @@ class DrinfeldModule(Parent, UniqueRepresentation):
sage: psi
Drinfeld module defined by T |--> (T + 1)*τ + T
sage: psi.base()
Fraction Field of Univariate Polynomial Ring in T over Finite Field in z2 of size 7^2 over its base
Fraction Field of Univariate Polynomial Ring in T over Finite Field in z2 of size 7^2

.. NOTE::

Expand Down Expand Up @@ -213,7 +213,7 @@ class DrinfeldModule(Parent, UniqueRepresentation):
:meth:`base`::

sage: phi.base()
Finite Field in z of size 3^12 over its base
Finite Field in z of size 3^12

The base morphism is retrieved using :meth:`base_morphism`::

Expand All @@ -223,14 +223,6 @@ class DrinfeldModule(Parent, UniqueRepresentation):
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
ring extension
(see :class:`sage.rings.ring_extension.RingExtension`) whose
underlying ring is `K` and whose base is the base morphism::

sage: phi.base() is K
False

.. RUBRIC:: Getters

One can retrieve basic properties::
Expand Down Expand Up @@ -403,7 +395,7 @@ class DrinfeldModule(Parent, UniqueRepresentation):

sage: action = phi.action()
sage: action
Action on Finite Field in z of size 3^12 over its base
Action on Finite Field in z of size 3^12
induced by Drinfeld module defined by T |--> τ^2 + τ + z

The action on elements is computed by calling the action object::
Expand Down Expand Up @@ -711,7 +703,7 @@ def __call__(self, a):
::

sage: a = A.random_element(5)
sage: phi(a)[0] == phi.category().base()(a)
sage: phi(a)[0] == phi.A_field()(a)
True
"""
return self._morphism(a)
Expand Down Expand Up @@ -859,7 +851,7 @@ def action(self):
sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
sage: action = phi.action()
sage: action
Action on Finite Field in z12 of size 5^12 over its base
Action on Finite Field in z12 of size 5^12
induced by Drinfeld module defined by T |--> z12^5*τ^2 + z12^3*τ + 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

Expand Down Expand Up @@ -1446,7 +1438,6 @@ def is_isomorphic(self, other, absolutely=False) -> bool:
if absolutely:
return True
else:
ue = ue.backend(force=True)
try:
_ = ue.nth_root(e)
except ValueError:
Expand Down
5 changes: 4 additions & 1 deletion src/sage/rings/ring_extension_element.pyx
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Expand Up @@ -162,12 +162,15 @@
sage: R.<x> = QQ[]
sage: E = R.over()
sage: P = E(x^2 + 2*x + 3)
sage: P(1)

Check failure on line 165 in src/sage/rings/ring_extension_element.pyx

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Failed example: 

Failed example:: Exception raised:
Traceback (most recent call last):
  File "/usr/share/miniconda/envs/sage-dev/lib/python3.12/site-packages/sage/doctest/forker.py", line 734, in _run
    self.compile_and_execute(example, compiler, test.globs)
  File "/usr/share/miniconda/envs/sage-dev/lib/python3.12/site-packages/sage/doctest/forker.py", line 1158, in compile_and_execute
    exec(compiled, globs)
  File "<doctest sage.rings.ring_extension_element.RingExtensionElement.__call__[3]>", line 1, in <module>
    P(Integer(1))
  File "sage/rings/ring_extension_element.pyx", line 171, in sage.rings.ring_extension_element.RingExtensionElement.__call__
    return from_backend(output, args[0].parent())
  File "sage/rings/ring_extension_conversion.pyx", line 457, in sage.rings.ring_extension_conversion.from_backend
    ans = from_backend_element(arg,E)
TypeError: Cannot convert sage.rings.integer_ring.IntegerRing_class to sage.rings.ring_extension.RingExtension_generic

Check failure on line 165 in src/sage/rings/ring_extension_element.pyx

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Failed example: 

Failed example:: Exception raised:
Traceback (most recent call last):
  File "/sage/src/sage/doctest/forker.py", line 734, in _run
    self.compile_and_execute(example, compiler, test.globs)
  File "/sage/src/sage/doctest/forker.py", line 1158, in compile_and_execute
    exec(compiled, globs)
  File "<doctest sage.rings.ring_extension_element.RingExtensionElement.__call__[3]>", line 1, in <module>
    P(Integer(1))
  File "sage/rings/ring_extension_element.pyx", line 171, in sage.rings.ring_extension_element.RingExtensionElement.__call__
  File "sage/rings/ring_extension_conversion.pyx", line 457, in sage.rings.ring_extension_conversion.from_backend
TypeError: Cannot convert sage.rings.integer_ring.IntegerRing_class to sage.rings.ring_extension.RingExtension_generic

Check failure on line 165 in src/sage/rings/ring_extension_element.pyx

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GitHub Actions / Conda (macos, Python 3.12, all)

Failed example: 

Failed example:: Exception raised:
Traceback (most recent call last):
  File "/Users/runner/miniconda3/envs/sage-dev/lib/python3.12/site-packages/sage/doctest/forker.py", line 734, in _run
    self.compile_and_execute(example, compiler, test.globs)
  File "/Users/runner/miniconda3/envs/sage-dev/lib/python3.12/site-packages/sage/doctest/forker.py", line 1158, in compile_and_execute
    exec(compiled, globs)
  File "<doctest sage.rings.ring_extension_element.RingExtensionElement.__call__[3]>", line 1, in <module>
    P(Integer(1))
  File "sage/rings/ring_extension_element.pyx", line 171, in sage.rings.ring_extension_element.RingExtensionElement.__call__
    return from_backend(output, args[0].parent())
  File "sage/rings/ring_extension_conversion.pyx", line 457, in sage.rings.ring_extension_conversion.from_backend
    ans = from_backend_element(arg,E)
TypeError: Cannot convert sage.rings.integer_ring.IntegerRing_class to sage.rings.ring_extension.RingExtension_generic

Check failure on line 165 in src/sage/rings/ring_extension_element.pyx

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GitHub Actions / Conda (macos, Python 3.13, all)

Failed example: 

Failed example:: Exception raised:
Traceback (most recent call last):
  File "/Users/runner/miniconda3/envs/sage-dev/lib/python3.13/site-packages/sage/doctest/forker.py", line 734, in _run
    self.compile_and_execute(example, compiler, test.globs)
    ~~~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/Users/runner/miniconda3/envs/sage-dev/lib/python3.13/site-packages/sage/doctest/forker.py", line 1158, in compile_and_execute
    exec(compiled, globs)
    ~~~~^^^^^^^^^^^^^^^^^
  File "<doctest sage.rings.ring_extension_element.RingExtensionElement.__call__[3]>", line 1, in <module>
    P(Integer(1))
    ~^^^^^^^^^^^^
  File "sage/rings/ring_extension_element.pyx", line 171, in sage.rings.ring_extension_element.RingExtensionElement.__call__
    return from_backend(output, args[0].parent())
  File "sage/rings/ring_extension_conversion.pyx", line 457, in sage.rings.ring_extension_conversion.from_backend
    ans = from_backend_element(arg,E)
TypeError: Cannot convert sage.rings.integer_ring.IntegerRing_class to sage.rings.ring_extension.RingExtension_generic

Check failure on line 165 in src/sage/rings/ring_extension_element.pyx

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GitHub Actions / Conda (ubuntu, Python 3.13, all)

Failed example: 

Failed example:: Exception raised:
Traceback (most recent call last):
  File "/usr/share/miniconda/envs/sage-dev/lib/python3.13/site-packages/sage/doctest/forker.py", line 734, in _run
    self.compile_and_execute(example, compiler, test.globs)
    ~~~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/share/miniconda/envs/sage-dev/lib/python3.13/site-packages/sage/doctest/forker.py", line 1158, in compile_and_execute
    exec(compiled, globs)
    ~~~~^^^^^^^^^^^^^^^^^
  File "<doctest sage.rings.ring_extension_element.RingExtensionElement.__call__[3]>", line 1, in <module>
    P(Integer(1))
    ~^^^^^^^^^^^^
  File "sage/rings/ring_extension_element.pyx", line 171, in sage.rings.ring_extension_element.RingExtensionElement.__call__
    return from_backend(output, args[0].parent())
  File "sage/rings/ring_extension_conversion.pyx", line 457, in sage.rings.ring_extension_conversion.from_backend
    ans = from_backend_element(arg,E)
TypeError: Cannot convert sage.rings.integer_ring.IntegerRing_class to sage.rings.ring_extension.RingExtension_generic

Check failure on line 165 in src/sage/rings/ring_extension_element.pyx

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GitHub Actions / Conda (ubuntu, Python 3.12, all)

Failed example: 

Failed example:: Exception raised:
Traceback (most recent call last):
  File "/usr/share/miniconda/envs/sage-dev/lib/python3.12/site-packages/sage/doctest/forker.py", line 734, in _run
    self.compile_and_execute(example, compiler, test.globs)
  File "/usr/share/miniconda/envs/sage-dev/lib/python3.12/site-packages/sage/doctest/forker.py", line 1158, in compile_and_execute
    exec(compiled, globs)
  File "<doctest sage.rings.ring_extension_element.RingExtensionElement.__call__[3]>", line 1, in <module>
    P(Integer(1))
  File "sage/rings/ring_extension_element.pyx", line 171, in sage.rings.ring_extension_element.RingExtensionElement.__call__
    return from_backend(output, args[0].parent())
  File "sage/rings/ring_extension_conversion.pyx", line 457, in sage.rings.ring_extension_conversion.from_backend
    ans = from_backend_element(arg,E)
TypeError: Cannot convert sage.rings.integer_ring.IntegerRing_class to sage.rings.ring_extension.RingExtension_generic

Check failure on line 165 in src/sage/rings/ring_extension_element.pyx

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GitHub Actions / Conda (ubuntu, Python 3.12, all, editable)

Failed example: 

Failed example:: Exception raised:
Traceback (most recent call last):
  File "/home/runner/work/sage/sage/src/sage/doctest/forker.py", line 734, in _run
    self.compile_and_execute(example, compiler, test.globs)
  File "/home/runner/work/sage/sage/src/sage/doctest/forker.py", line 1158, in compile_and_execute
    exec(compiled, globs)
  File "<doctest sage.rings.ring_extension_element.RingExtensionElement.__call__[3]>", line 1, in <module>
    P(Integer(1))
  File "sage/rings/ring_extension_element.pyx", line 171, in sage.rings.ring_extension_element.RingExtensionElement.__call__
  File "sage/rings/ring_extension_conversion.pyx", line 457, in sage.rings.ring_extension_conversion.from_backend
TypeError: Cannot convert sage.rings.integer_ring.IntegerRing_class to sage.rings.ring_extension.RingExtension_generic
6
"""
if (<RingExtension_generic>self._parent)._import_methods:
output = self._backend(*to_backend(args), **to_backend(kwargs))
return from_backend(output, self._parent)
if args:
return from_backend(output, args[0].parent())
else:
return from_backend(output, self._parent)
return TypeError("this element is not callable")

def __dir__(self):
Expand Down
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