fix coercion from libgap's finite fields, use libgap in sage/rings/finite_rings

[dimpase]

implement coercion from libgap's finite fields, to fix e.g.

sage: F=GF(25)
sage: F(libgap.Z(25)^3)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
...
TypeError: unable to coerce <class 'sage.libs.gap.element.GapElement_FiniteField'>

With the ticket branch, this works:

sage: F=GF(25)
sage: F(libgap.Z(25)^3)
4*z2 + 3

---

As well, we switch to use libgap instead of pexpect GAP - internally
in the affected files. This is a part of #26902

CC: @nbruin @mkoeppe @kwankyu

Component: number theory

Author: Dima Pasechnik

Branch/Commit: 6e5ea79

Reviewer: Matthias Koeppe

Issue created by migration from https://trac.sagemath.org/ticket/34770

[dimpase]

Description changed:

--- 
+++ 
@@ -12,8 +12,8 @@
 With the ticket branch, this works:
 
 ```
-sage: sage: F=GF(25)
-sage: sage: F(libgap.Z(25)^3)
+sage: F=GF(25)
+sage: F(libgap.Z(25)^3)
 4*z2 + 3
 ```
 

[dimpase]

Commit: 02bd544

[dimpase]

New commits:

02bd544

fix coersion of libgap FFelements; switch to libgap

[dimpase]

Branch: u/dimpase/rings/FFEs_to_accept_libgap

[dimpase]

comment:4

I've left intact the ability to accept pexpect GAP data to be converted in Sage finite field elements - this means I need a test for this type.
I'm using an import statement

from sage.interfaces.gap import is_GapElement

and a call to is_GapElement().
Is there a way to avoid this call? I can also put the import into the try block and
set is_GapElement() to always return False in case of import exception.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

21e38a7

fix coersion of libgap FFelements; switch to libgap

[sagetrac-git]

Changed commit from 02bd544 to 21e38a7

[sagetrac-git]

Changed commit from 21e38a7 to d4bde80

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d4bde80

make imports of is_GapEelment uniform across files

[dimpase]

comment:7

I can also resort to not calling is_GapElement(), and rely on exceptions - this would need a bit more rearranging of if/else.

However, there are still explicit tests like sage: S(gap('Z(25)^3')), and there is a discrepancy between libgap and (pexpect) gap in the way they treat such inputs:

sage: gap('Z(25)^3')
Z(5^2)^3
sage: gap.eval('Z(25)^3')
'Z(5^2)^3'
sage: libgap.eval('Z(25)^3')
Z(5^2)^3
sage: libgap('Z(25)^3')
"Z(25)^3"

So these tests will have to go as soon as gap is set to libgap.