Trac sagemath#34692: Frobenius endomorphism creation fail for infinit…

…e ring extension

I wish to instantiate the Frobenius endomorphism of the field `Fq(X)`
seen as a ring over `Fq[X]`. The following raises an exception:

{{{
sage: Fq = GF(11)
sage: FqX.<X> = Fq[]
sage: k = Frac(FqX)
sage: i = Hom(FqX, k).natural_map()
sage: K = k.over(i)
sage: K.frobenius_endomorphism()
------------------------------------------------------------------------
---
NotImplementedError                       Traceback (most recent call
last)
Input In [43], in <cell line: 6>()
      4 i = Hom(FqX, k).natural_map()
      5 K = k.over(i)
----> 6 K.frobenius_endomorphism()

File $SAGE_ROOT/sage/src/sage/rings/ring.pyx:1428, in
sage.rings.ring.CommutativeRing.frobenius_endomorphism()
   1426     """
   1427     from .morphism import FrobeniusEndomorphism_generic
-> 1428     return FrobeniusEndomorphism_generic(self, n)
   1429
   1430 def derivation_module(self, codomain=None, twist=None):

File $SAGE_ROOT/sage/src/sage/rings/morphism.pyx:2929, in
sage.rings.morphism.FrobeniusEndomorphism_generic.__init__()
   2927 if not isinstance(domain, CommutativeRing):
   2928     raise TypeError("The base ring must be a commutative ring")
-> 2929 self._p = domain.characteristic()
   2930 if not self._p.is_prime():
   2931     raise TypeError("the characteristic of the base ring must be
prime")

File $SAGE_ROOT/sage/src/sage/categories/rings.py:588, in
Rings.ParentMethods.characteristic(self)
    586 from sage.rings.infinity import infinity
    587 from sage.rings.integer_ring import ZZ
--> 588 order_1 = self.one().additive_order()
    589 return ZZ.zero() if order_1 is infinity else order_1

File $SAGE_ROOT/sage/src/sage/rings/ring_extension_element.pyx:415, in
sage.rings.ring_extension_element.RingExtensionElement.additive_order()
    413         5
    414     """
--> 415     return self._backend.additive_order()
    416
    417 def multiplicative_order(self):

File $SAGE_ROOT/sage/src/sage/structure/element.pyx:2818, in
sage.structure.element.RingElement.additive_order()
   2816     Return the additive order of ``self``.
   2817     """
-> 2818     raise NotImplementedError
   2819
   2820 def multiplicative_order(self):

NotImplementedError:
}}}

Best,

Antoine Leudière

URL: https://trac.sagemath.org/34692
Reported by: antoine-leudiere
Ticket author(s): Antoine Leudière
Reviewer(s): Xavier Caruso, David Ayotte

src/sage/rings/ring_extension.pyx

@@ -1885,6 +1885,59 @@ cdef class RingExtension_generic(CommutativeAlgebra):
         parent = self.Hom(codomain, category=category)
         return RingExtensionHomomorphism(parent, im_gens, base_map, check)
 
+    def characteristic(self):
+        r"""
+        Return the characteristic of the extension as a ring.
+
+        OUTPUT:
+
+        A prime number or zero.
+
+        EXAMPLES::
+
+            sage: F = GF(5^2).over()   # over GF(5)
+            sage: K = GF(5^4).over(F)
+            sage: L = GF(5^12).over(K)
+            sage: F.characteristic()
+            5
+            sage: K.characteristic()
+            5
+            sage: L.characteristic()
+            5
+
+        ::
+
+            sage: F = RR.over(ZZ)
+            sage: F.characteristic()
+            0
+
+        ::
+
+            sage: F = GF(11)
+            sage: A.<x> = F[]
+            sage: K = Frac(F).over(F)
+            sage: K.characteristic()
+            11
+
+        ::
+
+            sage: E = GF(7).over(ZZ)
+            sage: E.characteristic()
+            7
+
+        TESTS:
+
+            Ensure ticket :trac:`34692` is fixed::
+
+            sage: Fq = GF(11)
+            sage: FqX.<X> = Fq[]
+            sage: k = Frac(FqX)
+            sage: K = k.over(FqX)
+            sage: K.frobenius_endomorphism()
+            Frobenius endomorphism x |--> x^11 of Fraction Field of Univariate Polynomial Ring in X over Finite Field of size 11 over its base
+        """
+        return self._backend.characteristic()
+
 
 # Fraction fields
 #################