p-adic fields should have Witt Frobenius

[dmharvey]

If K is an unramified extension of Qp, there should be a function that computes the canonical Witt p-Frobenius:

sage: K.<a> = Qp(25)
sage: a.witt_frobenius()
???

---

Apply trac_8241-frobenius.patch​ to the Sage library.

[dmharvey]

(oops the Qp above should be Qq)

[roed]

I called the function frobenius rather than witt_frobenius.

[roed]

Fixed a small problem revealed by the test-bot.

[kedlaya]

For consistency, these methods should also apply to Qp, with Frobenius acting as the identity map. For instance, the following should not raise an exception:

sage: Qp(7)(2).frobenius()

[roed]

There's now a frobenius method for p-adic base rings and fields.

[roed]

I have no idea why adding a method to Qp causes a test in sage/libs/fplll/fplll.py to fail.

[jdemeyer]

Replying to roed:

I have no idea why adding a method to Qp causes a test in sage/libs/fplll/fplll.py to fail.

See #10195.

[leif]

Ping. (#11586, which has positive review, depends on this one.)

[roed]

As Sebastian Pancratz noted in his talk, it would be great if someone wrote a better implementation of this.

[davidloeffler]

The patchbot finds some Sphinx formatting errors in the docstrings:

docstring of sage.rings.padics.padic_ZZ_pX_CA_element.pAdicZZpXCAElement.teichmuller_list:4: WARNING: Inline interpreted text or phrase reference start-string without end-string.
docstring of sage.rings.padics.padic_ZZ_pX_CA_element:2: WARNING: Block quote ends without a blank line; unexpected unindent.
docstring of sage.rings.padics.padic_ZZ_pX_CR_element.pAdicZZpXCRElement.teichmuller_list:4: WARNING: Inline interpreted text or phrase reference start-string without end-string.
docstring of sage.rings.padics.padic_ZZ_pX_CR_element:2: WARNING: Block quote ends without a blank line; unexpected unindent.
docstring of sage.rings.padics.padic_ext_element.pAdicExtElement.frobenius:6: WARNING: Inline interpreted text or phrase reference start-string without end-string.

It's also harping on about trailing whitespace (which is apparently now a Bad Thing, although Sage has been tolerating it cheerfully for years).

[davidloeffler]

Here's a new patch with the docstrings straightened out and trailing whitespace removed. I've also added a doctest to show that an error is raised when "frobenius" is called on an element of a ramified extension. I'm happy with the rest of the code, so if David's happy with my changes we can call this that's a positive review.

[roed]

Looks good to me. I've created #12657: write a more efficient implementation of Frobenius.

[davidloeffler]

Apply only this patch. Patch against 5.0.beta7

[davidloeffler]

Patchbot's grumbling about trailing whitespace again, so I removed one single space character and re-uploaded the patch.