Fix finite field modulus handling

[jdemeyer]

For prime finite fields, we should support a custom modulus (internally in Sage, we don't need to use this modulus in the backend):

sage: x = polygen(GF(7))
sage: k = GF(7, modulus=x+5)
sage: k.modulus()
x + 6
sage: k.gen()
1

We should also make the modulus monic:

sage: x = polygen(GF(7))
sage: k.<a> = GF(7^2, modulus=2*x^2-3, impl="pari_ffelt")
sage: k.modulus()
2*x^2 + 4

(the Givaro backend does this implicitly)

The patch does the above, with lots of cleaning up.

Depends on #16934

CC: @pjbruin @jpflori

Component: finite rings

Author: Jeroen Demeyer

Branch/Commit: 0d9b5cc

Reviewer: Jean-Pierre Flori

Issue created by migration from https://trac.sagemath.org/ticket/16983

[jdemeyer]

Dependencies: #16934

[jdemeyer]

Author: Jeroen Demeyer

[jdemeyer]

Branch: u/jdemeyer/ticket/16983

[jdemeyer]

New commits:

8f805a2	Trac 16934: always put finite field implementation in factory key
8923777	Trac 16934: remove method FiniteFieldFactory.other_keys()
5e2681e	Trac 16934: more cleaning up
ffb74fa	Trac 16934: add doctest
33c86ef	Merge remote-tracking branch 'origin/develop' into ticket/16934
bf36ec1	Trac 16934: put back (temporary?) warning about modulus parameter
9cbdcde	Trac 16934: fix warning about ignored 'modulus' argument
eb5553e	Trac #16983: no longer ignore modulus for prime finite fields

[jdemeyer]

Commit: eb5553e

[sagetrac-git]

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

b6a6c66

Trivial changes to the FiniteField_pari_ffelt constructor

[sagetrac-git]

Changed commit from eb5553e to b6a6c66

[jpflori]

comment:9

I get a failing doctest in src/sage/rings/finite_rings/homset.py because some ordering changed.

[sagetrac-git]

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

f7eccdc

Merge remote-tracking branch 'origin/develop' into ticket/16983

[sagetrac-git]

Changed commit from b6a6c66 to f7eccdc

[sagetrac-git]

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

dd16a75

Fix order in doctest output

[sagetrac-git]

Changed commit from f7eccdc to dd16a75

[sagetrac-git]

Changed commit from dd16a75 to 5028e10

[sagetrac-git]

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

5028e10

Fix infinite loop in unpickling

[jdemeyer]

New commits:

5028e10

Fix infinite loop in unpickling

[jpflori]

comment:15

There is something missing here:

+ def polynomial(self, name=None):
+ """
+ Return the irreducible characteristic polynomial of the
+ generator of this finite field, i.e., the polynomial `f(x)` so
+ elements of the finite field as elements modulo `f`.

[jpflori]

comment:16

Otherwise, this looks fine to me and all tests pass.
I'll give a final look a little bit later this afternoon.

[sagetrac-git]

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

0d9b5cc

Clarify docs for polynomial()

[sagetrac-git]

Changed commit from 5028e10 to 0d9b5cc

[jdemeyer]

comment:18

Replying to @jpflori:

There is something missing here:

+ def polynomial(self, name=None):
+ """
+ Return the irreducible characteristic polynomial of the
+ generator of this finite field, i.e., the polynomial `f(x)` so
+ elements of the finite field as elements modulo `f`.

I just moved that sentence, in fact it dates back to

commit a4928b0cf450b765c3bde6e5328f0146603526a8
Author: tornaria <tornaria@math.utexas.edu>
Date:   Sat Feb 11 01:13:08 2006 +0000

    [project @ original sage-0.10.12]

---

New commits:

0d9b5cc

Clarify docs for polynomial()

[jpflori]

Reviewer: Jean-Pierre Flori

[jpflori]

comment:19

Ok, looks good to me.

[vbraun]

Changed branch from u/jdemeyer/ticket/16983 to 0d9b5cc