Frobenius endomorphism over p-adics

[xcaruso]

This patch provides a method frobenius_endomorphism which returns the Frobenius endomorphism of a p-adic field.

Here is a small demo:

sage: K.<a> = Qq(5^3)
sage: Frob = K.frobenius_endomorphism()

sage: Frob
Frobenius endomorphism on Unramified Extension of 5-adic ... lifting a |--> a^5 on the residue field
sage: Frob^12
Identity endomorphism of Unramified Extension of 5-adic Field ...

sage: Frob(a) = a.frobenius()
True

Component: padics

Keywords: Frobenius

Author: Xavier Caruso

Branch/Commit: u/kedlaya/ticket/14824 @ dc5b393

Reviewer: Kiran Kedlaya

Issue created by migration from https://trac.sagemath.org/ticket/14824

[roed314]

comment:2

• Why do you need to store _degree_fixed?
• There are a couple functions without doctests, and the implementation of hash can be easily improved (use power rather than the string representation).
• One doctest failure:

File "/mnt/storage2TB/patchbot/Sage/sage-5.11.beta3/devel/sage/sage/rings/padics/padic_generic.py", line 566, in sage.rings.padics.padic_generic.pAdicGeneric.frobenius_endomorphism
Failed example:
    K.frobenius_endomorphism(6) == Frob
Expected:
    True
Got:
    False

[xcaruso]

comment:3

Replying to @roed314:

• Why do you need to store _degree_fixed?

Oops. It was because I was first planning to implement a method fixed_points but then I gave up. It seems that I forgot to remode _degree_fixed.

• There are a couple functions without doctests, and the implementation of hash can be easily improved (use power rather than the string representation).

Done.

• One doctest failure:

Fixed.

[xcaruso]

Attachment: trac_14824_frobenius_endomorphism_padics.patch.gz

[roed314]

comment:4

Note that this ticket does not resolve the horrible performance of the frobenius method: see #12657. But the changes here are certainly worth incorporating, and don't need to wait on #12657.

[kedlaya]

Branch: u/kedlaya/ticket/14824

[sagetrac-git]

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

dc5b393

Merge branch 'develop' into ticket/14824

[sagetrac-git]

Commit: dc5b393

[kedlaya]

comment:7

With help from roed, I converted this to the git workflow; it merges into 6.1beta3 with only trivial modifications. Looks good to me.

[kedlaya]

Reviewer: Kiran Kedlaya