#17516: Radical expressions for roots of polynomials in more cases
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Reporter: gagern | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.5
Component: number fields | Keywords: radical galois symbolic
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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Given a polynomial from ℚ[X], we need a better way to express its roots
using radical expressions if such an expression is possible.
The current approach, as used by e.g. `NumberFieldElement._symbolic_` (and
after #14239 gets merged probably
`AlgebraicNumber_base.radical_expression` instead), delegates this task to
the `solve` method for expressions from the symbolic ring. That in turn
will delegate to Maxima. But Maxima is not able to find a radical
expression in all cases where they do exist:
{{{
sage: p = x^6-300*x^5+30361*x^4-1061610*x^3+1141893*x^2-915320*x+101724
sage: p.solve(x, explicit_solutions=True)
[]
sage: r = 1/8*((sqrt(4*(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) -
4/3/(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) + 17) + 5)^2 +
4)*(sqrt(4*(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) - 4/3/(1/9*sqrt(109)*sqrt(3)
+ 2)^(1/3) + 17) + 5)
sage: r.minpoly() == QQ[x](p)
True
}}}
In comment:77:ticket:14239 Jeroen Demeyer stated that a proper solution
here would use Galois Theory, and that we might be able to leverage PARI
for this. So the goal of this ticket here is a function or method which
constructs radical expressions for the roots of all polynomials where
doing so is possible, perhaps with some explicitely stated bound on the
degree.
--
Ticket URL: <http://trac.sagemath.org/ticket/17516>
Sage <http://www.sagemath.org>
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