#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 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 = K.galois_closure(names="b")
sage: q = GC.defining_polynomial()
sage: gal = pari(q).galoisinit()
sage: G = PermutationGroup(sorted(gal[6], cmp=cmp))
sage: G.is_solvable()
True
sage: ds = G.derived_series()
}}}
I still have to figure out how to turn that derived series into a radical
expression. I hope you don't mind me posting thoughts along the way.
--
Ticket URL: <http://trac.sagemath.org/ticket/17516#comment:3>
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