From 501a25411904ead8d9b097f465cd40b984eaa701 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Antoine=20Leudi=C3=A8re?= <antoine.leudiere@inria.fr>
Date: Mon, 14 Nov 2022 16:20:02 +0100
Subject: [PATCH 1/4] Implement `characteristic` method for rings extensions

See ticket #34692.
---
 src/sage/rings/ring_extension.pyx | 42 +++++++++++++++++++++++++++++++
 1 file changed, 42 insertions(+)

diff --git a/src/sage/rings/ring_extension.pyx b/src/sage/rings/ring_extension.pyx
index 2b649359ad4..9f53d42f80f 100644
--- a/src/sage/rings/ring_extension.pyx
+++ b/src/sage/rings/ring_extension.pyx
@@ -1885,6 +1885,48 @@ 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.
+
+        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: R = Zp(7).over(ZZ)
+            sage: R.characteristic()
+            0
+        """
+        return self._backend.characteristic()
+
 
 # Fraction fields
 #################

From bcffa0cf0842ebef2403d5c8b6aa88514beb86e1 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Antoine=20Leudi=C3=A8re?= <antoine.leudiere@inria.fr>
Date: Mon, 14 Nov 2022 17:18:21 +0100
Subject: [PATCH 2/4] Fix imprecision in documentation

See David's comment: https://trac.sagemath.org/ticket/34692#comment:5.
---
 src/sage/rings/ring_extension.pyx | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/rings/ring_extension.pyx b/src/sage/rings/ring_extension.pyx
index 9f53d42f80f..bbc082d2e77 100644
--- a/src/sage/rings/ring_extension.pyx
+++ b/src/sage/rings/ring_extension.pyx
@@ -1891,7 +1891,7 @@ cdef class RingExtension_generic(CommutativeAlgebra):
 
         OUTPUT:
 
-        A prime number.
+        A prime number or zero.
 
         EXAMPLES::
 

From 675e24988a07f58d092a7ddb976c9e6ee04f3ba6 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Antoine=20Leudi=C3=A8re?= <antoine.leudiere@inria.fr>
Date: Mon, 14 Nov 2022 18:03:12 +0100
Subject: [PATCH 3/4] Add tests for characteristic method

See https://trac.sagemath.org/ticket/34692#comment:9.
---
 src/sage/rings/ring_extension.pyx | 18 +++++++++++++++---
 1 file changed, 15 insertions(+), 3 deletions(-)

diff --git a/src/sage/rings/ring_extension.pyx b/src/sage/rings/ring_extension.pyx
index bbc082d2e77..c761314585c 100644
--- a/src/sage/rings/ring_extension.pyx
+++ b/src/sage/rings/ring_extension.pyx
@@ -1921,9 +1921,21 @@ cdef class RingExtension_generic(CommutativeAlgebra):
 
         ::
 
-            sage: R = Zp(7).over(ZZ)
-            sage: R.characteristic()
-            0
+            sage: E = Fp(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: i = Hom(FqX, k).natural_map()
+            sage: K = k.over(i)
+            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()
 

From ab2c211033e15fa5237b20690a715d50577d06f1 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Antoine=20Leudi=C3=A8re?= <antoine.leudiere@inria.fr>
Date: Mon, 14 Nov 2022 18:52:42 +0100
Subject: [PATCH 4/4] Fix doctest

See https://trac.sagemath.org/ticket/34692#comment:11.
---
 src/sage/rings/ring_extension.pyx | 9 ++++-----
 1 file changed, 4 insertions(+), 5 deletions(-)

diff --git a/src/sage/rings/ring_extension.pyx b/src/sage/rings/ring_extension.pyx
index c761314585c..b493f57bed9 100644
--- a/src/sage/rings/ring_extension.pyx
+++ b/src/sage/rings/ring_extension.pyx
@@ -1921,19 +1921,18 @@ cdef class RingExtension_generic(CommutativeAlgebra):
 
         ::
 
-            sage: E = Fp(7).over(ZZ)
+            sage: E = GF(7).over(ZZ)
             sage: E.characteristic()
             7
 
-        TESTS::
+        TESTS:
 
-            Ensure ticket :trac:`34692` is fixed.
+            Ensure ticket :trac:`34692` is fixed::
 
             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 = 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
         """