finite-field .extension() sets up wrong coercion

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Failing example:

K.<u> = GF(23^4)
L.<v> = K.extension(5)
assert L(u).minpoly() == u.minpoly()

This is because K uses a Conway polynomial, but L doesn't, so the following code in FiniteField.extension() is wrong (introduced in #24195):

                if self.is_conway(): # and E is Conway
                    alpha = E.gen()**((E.order()-1)//(self.order()-1))
                else:
                    alpha = self.modulus().any_root(E)
                try: # to register a coercion map (embedding of self to E)
                    E.register_coercion(self.hom([alpha], codomain=E, check=False))

The mysterious comment "# and E is Conway" tells us exactly what to do to fix the bug.

CC: @kwankyu @videlec

Component: finite rings

Author: Lorenz Panny

Branch/Commit: 3b6be5f

Reviewer: Kwankyu Lee, Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/33937

[yyyyx4]

Description changed:

--- 
+++ 
@@ -1,16 +1,8 @@
 Failing example:
 
 ```sage
-T = polygen(GF(23))
-F.<t> = GF(23^4, modulus=T^4+T+2)
-
-x = polygen(F)
-f = x^3 - t
-K.<u> = f.splitting_field()
-
-y = polygen(K)
-g = y^2 - u
-L.<v> = g.splitting_field()
-
+K.<u> = GF(23^4)
+L.<v> = K.extension(5)
 assert L(u).minpoly() == u.minpoly()
 ```
+

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Author: Lorenz Panny

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Commit: 7cf3cf3

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Branch: public/33937

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New commits:

7cf3cf3

fix computation of finite-field coercion (Conway into non-Conway)

[yyyyx4]

Description changed:

--- 
+++ 
@@ -6,3 +6,15 @@
 assert L(u).minpoly() == u.minpoly()
 ```
 
+This is because `K` uses a Conway polynomial, but `L` doesn't, so the following code in `FiniteField.extension()` is wrong (introduced in #24195):
+
+```python
+                if self.is_conway(): # and E is Conway
+                    alpha = E.gen()**((E.order()-1)//(self.order()-1))
+                else:
+                    alpha = self.modulus().any_root(E)
+                try: # to register a coercion map (embedding of self to E)
+                    E.register_coercion(self.hom([alpha], codomain=E, check=False))
+```
+
+The mysterious comment "# and E is Conway" tells us exactly what to do to fix the bug.

[fchapoton]

comment:3

Maybe use the new syntax

sage: GF((13,11))
Finite Field in z11 of size 13^11

in the doctests ?