#5159: Add functionality to Galois groups
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Reporter: davidloeffler | Owner: davidloeffler
Type: enhancement | Status: assigned
Priority: major | Milestone: sage-3.4
Component: number theory | Resolution:
Keywords: galois groups, number theory |
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Changes (by davidloeffler):
* status: new => assigned
Comment:
The above patch should do the job for absolute fields: it creates a new
GaloisGroup class, which has very little resemblance to the old one, and
derives from PermutationGroup_generic. The init script calls Pari's
galoisinit. Elements are stored as GaloisGroupElement objects, which are
basically permutations, but with the addition of a cached method that
returns the image of a generator of that permutation under the
corresponding automorphism.
I've also added toy implementations of decomposition and ramification
groups and the Artin symbol for prime ideals in number fields; I'm sure
there are faster algorithms to calculate these rather than using the
definitions directly as I've done, but I am a strong believer in toy
implementations.
At present this is all only for absolute fields, because Pari has no
direct support for Galois groups of relative extensions. Relative fields
could also be implemented, at least when the corresponding absolute field
is Galois over QQ, by calculating the Galois group of the absolute
extension and checking which automorphisms fix the intermediate field; but
I haven't done this.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5159#comment:1>
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