#2217: splitting field function for number fields
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Reporter: jason | Owner: davidloeffler
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.8
Component: number fields | Keywords:
Work_issues: | Upstream: N/A
Reviewer: | Author:
Merged: | Dependencies: #11904, #11995
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Description changed by jdemeyer:
Old description:
> {{{
> I agree that this would be a useful funtion to have. I would call it
> splitting_field() with a description similar to that of root_field()
>
> ...
>
> In the meantim you should be able to work with what is available as
> follows:
>
> sage: QQx.<x>=QQ[]
> sage: f=(x^2-2)*(x^2-3)
> sage: F=NumberField([p for p,n in f.factor()],'a')
> sage: F2=F.absolute_field('b')
> sage: F2.structure()
>
> (Isomorphism from Number Field in b with defining polynomial x^4 -
> 10*x^2 + 1 to Number Field in a0 with defining polynomial x^2 - 3 over
> its base field,
> Isomorphism from Number Field in a0 with defining polynomial x^2 - 3
> over its base field to Number Field in b with defining polynomial x^4
> - 10*x^2 + 1)
>
> Here F is first defined as a relative extension, with generators a0,a1
> satisfying the equations:
>
> sage: a0,a1=F.gens()
> sage: a0^2, a1^2
> (3, 2)
>
> then F2 is the associated absolute field, with F2.structure() giving
> maps from each of these into the other.
>
> sage: F2toF, FtoF2=F2.structure()
> sage: FtoF2(a0)
> -1/2*b^3 + 11/2*b
> sage: FtoF2(a0).minpoly()
> x^2 - 3
> sage: FtoF2(a1)
> -1/2*b^3 + 9/2*b
> sage: FtoF2(a1).minpoly()
> x^2 - 2
>
> }}}
>
> See the thread at http://groups.google.com/group/sage-
> devel/browse_thread/thread/32fe12de12d5f6a5/c91753b5e65fe7b9#c91753b5e65fe7b9
New description:
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
--
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/2217#comment:4>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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