Add functionality for S-units to UnitGroup

[JohnCremona]

In Sage-5.11, there is a lot of functionality for computing with S-units (see #14746) but no support for this in the UnitGroup class in sage/rings/number_field/unit_group.py. This ticket adds such support, also adding a wrapper for the PARI function bnfissunit.

Apply: attachment: trac_14963-Sunits.patch, attachment: trac_14963-reviewer.patch

Depends on #14489
Depends on #14746
Depends on #14519

CC: @sagetrac-mkosters @sagetrac-akoutsianas

Component: number fields

Keywords: sd51, units, S-units

Author: John Cremona

Reviewer: Peter Bruin

Merged: sage-5.13.beta0

Issue created by migration from https://trac.sagemath.org/ticket/14963

[JohnCremona]

Dependencies: #14489, #14746

[JohnCremona]

Changed keywords from units, S-units to sd51, units, S-units

[JohnCremona]

comment:4

I started to work on this at sd51 and hope to finish it soon.

[JohnCremona]

comment:5

It was quite easy to extend the functionality of the UnitGroup class to include S-units, thanks to the pari function bnfsunit (which was already wrapped) and bnfissunit (which was not). The class constructor requires that S not be a list, but it can be a tuple or something from which an element of the number field can be constructed. Note that the pari function bnfsunit only returns the S-unit generators which are not global units, and the bnfissunit returns the exponents in the order (1) global units of infinite order, (2) torsion, (3) other S-units; so some permutation is required. The new method for number fields is S_unit_group and is simiar to unit_group: it caches S-unit groups using the tuple S as key, with separate caches for proof and non-proof options.

The old number field method S_units is probably now redundant; I added a note there directing users to S_unit_group for great functionality, as we do for units.

[JohnCremona]

Author: John Cremona

[pjbruin]

Description changed:

--- 
+++ 
@@ -1 +1 @@
-In Sage-5.11, there is a lot of functionality for computing with S-units (see #14746) but no class for these which generalises the UnitGroup class in sage/rings/number_field/unit_group.py.   This ticket will create such a class, also adding a wrapper for the pari function bnfissunit.
+In Sage-5.11, there is a lot of functionality for computing with S-units (see #14746) but no support for this in the `UnitGroup` class in `sage/rings/number_field/unit_group.py`.   This ticket creates such functionality, also adding a wrapper for the PARI function `bnfissunit`.

[pjbruin]

comment:8

update summary and description