splitting field function for number fields

[jason]

We should add splitting_field() function: given a polynomial, compute the number field generated by all the roots.

See the thread at  http://groups.google.com/group/sage-devel/browse_thread/thread/32fe12de12d5f6a5/c91753b5e65fe7b9#c91753b5e65fe7b9

[cwitty]

Note that this approach does not give the splitting field. It gives a field containing at least one root of each factor of the original polynomial, but that still might not be the splitting field.

Later in the thread mentioned above, I give a technique using internals of qqbar which I believe does give the splitting field (perhaps inefficiently).

[jdemeyer]

Add splitting_field() function

[roed]

Jeroen, what is the status of the patch here?

[jdemeyer]

I totally forgot about this. I might be good to revisit this.

[jdemeyer]

I originally had plans for some speed-ups, but since the code works fine, I guess it can be reviewed.

[mmanes]

I ran all standard tests, and everything passed.

I was trying to test functionality, but I'm confused by the differences file. All of the old examples seem to work, and none of the new ones work.

In the first example, I get:

sage: G = NumberField(x^3 - x - 1,'a').galois_closure('b').galois_group(); G

Galois group of Number Field in b with defining polynomial x^6 - 14*x^4
+ 20*x^3 + 49*x^2 - 140*x + 307

The expected output seems to have been changed from this result to

Number Field in b with defining polynomial x^6 - 6*x^4 + 9*x^2 + 23 

These fields are isomorphic, but I've tried the example on three machines, and all of them give the first thing as the output.

Similarly, the second example doesn't work:

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

Traceback (click to the left of this block for traceback)
...
TypeError: permutation [(1, 2, 3), (4, 5, 6)] not in Galois group of
Number Field in b with defining polynomial x^6 - 14*x^4 + 20*x^3 +
49*x^2 - 140*x + 307

But the original example (now deleted) does work:

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

Subgroup [(), (1,5,2)(3,4,6), (1,2,5)(3,6,4)] of Galois group of Number
Field in b with defining polynomial x^6 - 14*x^4 + 20*x^3 + 49*x^2 -
140*x + 307