S-units sometimes broken and sometimes just plain wrong for relative fields

[davidloeffler]

The code for S-unit groups of number fields calls the degree method. For relative number fields this deliberately returns an error, because of the ambiguity between absolute and relative degree.

sage: L.<a,b> = NumberField([x^2 + 1, x^2 - 5])
sage: sage: p = L.ideal((-1/2*b - 1/2)*a + 1/2*b - 1/2)
sage: sage: W = L.S_units([p]); W
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
...
NotImplementedError: For a relative number field you must use relative_degree or absolute_degree as appropriate

In this case I think it should be absolute_degree, but changing this returns wrong output:

sage: L.<a,b> = NumberField([x^2 + 1, x^2 - 5])
sage: p = L.ideal((-1/2*b - 1/2)*a + 1/2*b - 1/2)
sage: p.absolute_norm()
9
sage: p.is_prime()
True
sage: W = L.S_units([p]); W
[1/2*a + 7/4, a, 1/2*b - 1/2]
sage: W[0].valuation(L.primes_above(2)[0])
-4

So the first element of the list of S-units isn't actually an S-unit!

In other examples the code just blows up, because it calls residue_field and that dies because of #8721:

sage: L.<a, b> = NumberField([polygen(QQ)^2 - 3, polygen(QQ)^2 - 5])
sage: L.S_units([L.ideal(a)])

This is arguably less bad: raising an error is far better than silently a wrong answer.

[davidloeffler]

Here's a patch. Turns out that the code was using K.gen and the correct answer is to call K.absolute_generator, which isn't the same in the above example. This fixes the first example; the second is an instance of #8721.

[davidloeffler]

apply over patches at #8446

[cremona]

Looks good, applied fine to 4.4.alpha0 + #8446 patches, and all tests in sage/rings/number_field pass.

Positive review!

[jhpalmieri]

Merged into 4.4.alpha2.