On Sat, Aug 23, 2014 at 8:49 AM, jonhanke <jonha...@gmail.com> wrote:
> Dear Sage-support,
>
> I'd like to use cloud.sagemath.com to compute the subfields F of the Hilbert
> class field of QQ(sqrt(-39)), which is a Galois extension of QQ with Galois
> group D_8. For number fields defined by a single polynomial it seems (from
> the documentation here) we can say
>
> L.<a> = NumberField(x^4+1)
> G = L.galois_group()
> H = G.decomposition_group(L.primes_above(3)[0])
> F = H.fixed_field()[0]
>
>
> to produce a number field F associated to the subgroup H of G. This works
> well, but how can I run over all subgroups of G? The desired command
>
> G.subgroups()
>
>
> we would like seems to require that G is a PermuatationGroup(), but G has
> type
>
> type(G)
>
>
> <class 'sage.rings.number_field.galois_group.GaloisGroup_v2_with_category'>
>
>
> Question 1: How can we coerce between these types of groups to be able to
> enumerate subgroups?
>
> If one tries to use the hilbert_class_field() command then one gets a
> relative extension
>
> K = NumberField(x^2+39,'a')
>
> K.class_group()
>
> Class group of order 4 with structure C4 of Number Field in a with defining
> polynomial x^2 + 39
>
> H = K.hilbert_class_field('b'); H
> Number Field in b with defining polynomial x^4 + 2*x^3 + 2*x^2 + x + 1 over
> its base field
>
This doesn't answer any of your questions below, but it answers the
subject line of your email:
L.<c> = H.absolute_field()
L.subfields()
>
> but the type of the Galois group one gets from this construction is not as
> nice
>
> G = H.galois_group()
> type(G)
> <class 'sage.rings.number_field.galois_group.GaloisGroup_v1'>
>
>
>
> and in particular this Galois group type does not support decomposition
> groups. A similar thing happens when one considers a tower of fields
>
> k.<a,b,c> = NumberField([x^2 + 1, x^2 + 3, x^2 + 5])
> GG = k.galois_group()
> type(GG)
> <class 'sage.rings.number_field.galois_group.GaloisGroup_v1>
>
>
> This leads to...
>
> Question 2: (How) Can we access the more robust Galois group type for
> absolute number fields made as a tower of fields?
>
> Any comments are appreciated. Thanks a lot!
>
> -Jon
> =)
>
>
>
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--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org
wst...@uw.edu
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