Specify subgroups of Galois groups using generators

[pbruin]

It should be possible to construct a subgroup of the Galois group of a number field from a set of generators, just as for permutation groups. This is already supported to some extent, but the list of elements of such a subgroup is wrong.

sage: R.<x> = QQ[]
sage: f = x^6 - 6*x^4 + 9*x^2 + 23
sage: K.<a> = NumberField(f)
sage: G = K.galois_group()
sage: list(G)
[(),
 (1,2,3)(4,5,6),
 (1,3,2)(4,6,5),
 (1,4)(2,6)(3,5),
 (1,5)(2,4)(3,6),
 (1,6)(2,5)(3,4)]
sage: g = G[1]
sage: h = G[3]

Before:

sage: list(G.subgroup([]))
[]
sage: list(G.subgroup([g]))
[(1,2,3)(4,5,6)]
sage: list(G.subgroup([h]))
[(1,4)(2,6)(3,5)]
sage: list(G.subgroup([g,h]))
[(1,2,3)(4,5,6), (1,4)(2,6)(3,5)]

After:

sage: list(G.subgroup([]))
[()]
sage: list(G.subgroup([g]))
[(), (1,2,3)(4,5,6), (1,3,2)(4,6,5)]
sage: list(G.subgroup([h]))
[(), (1,4)(2,6)(3,5)]
sage: list(G.subgroup([g,h]))
[(),
 (1,2,3)(4,5,6),
 (1,3,2)(4,6,5),
 (1,4)(2,6)(3,5),
 (1,5)(2,4)(3,6),
 (1,6)(2,5)(3,4)]

There is one user-visible change: the special case where applying subgroup() to the list of all elements of G returns G itself has been removed. Besides simplifying the code, this has the advantage that the object returned by subgroup() now always has type GaloisGroup_subgroup, and in particular has a method fixed_field().

[git]

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

​1c199f4

Merge branch 'develop' into ticket/26816-GaloisGroup_subgroup

[roed]

Looks good to me. Recent patchbot failures due to transition to Python 3.