#17516: Radical expressions for roots of polynomials using Galois theory
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Reporter: gagern | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.5
Component: number fields | Resolution:
Keywords: radical galois symbolic | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: #14239 | Stopgaps:
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Comment (by jdemeyer):
Replying to [comment:3 gagern]:
> Computing the Galois group for a polynomial works pretty fast, but I'm
not sure how much use that really is. I fear we might need the map between
the original polynomial and the Galois closure. Converting the Galois
closure for the example above takes some time, but not so much as
converting that closure to pari. The time is apparently spent somewhere
inside `_pari_integral_basis`. Now I wonder, do we actually have to call
`galoisinit` on the galois closure as a number field? Or could we do that
call on its defining polynomial instead? Something along these lines:
>
> {{{
> sage: Qx.<x> = QQ[]
> sage: p = x^6-300*x^5+30361*x^4-1061610*x^3+1141893*x^2-915320*x+101724
> sage: K = NumberField(p, names="a")
> sage: GC, GCm = K.galois_closure(names="b", map=True)
> sage: q = GC.defining_polynomial()
> sage: gal = pari(q).galoisinit()
> }}}
Yes indeed
> {{{
> sage: G = PermutationGroup(sorted(gal[6], cmp=cmp))
> sage: G.is_solvable()
> True
> sage: ds = G.derived_series()
> }}}
I don't think need you need to convert the group to Sage, I would use
`galoisfixfield()` from PARI. The tricky part will be adding the roots of
unity, you need to add them manually.
--
Ticket URL: <http://trac.sagemath.org/ticket/17516#comment:4>
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