#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):
I still have trouble figuring out what element we should adjoin. One
option
([http://en.wikipedia.org/wiki/Resolvent_%28Galois_theory%29#Terminology
Lagrange resolvent]?) would apparently be sum(x,,i,,z^i^ for in in
range(n))^n^ where the x,,i,, are the roots of the polynomial and z would
be a primitive n-th root of unity. For that we'd need to know the roots
and the cyclic order for them. It's easy to compute the roots in Sage but
they don't come with the cyclic structure. On the other hand, PARI
computes the roots and gives the automorphisms in terms of these, but its
roots are modulo p^e^, so they contain more information than GF(p) but I
don't know how to match them to the algebraic roots in Sage, or how to
perform a computation on them and turn the result into a number field
element. I'm still not sure how to do the recusrion once this is resolved,
but at the moment, I'm really wondering whether any of you knows how to
match these roots.
Alternative approaches to [http://math.stackexchange.com/q/1077722/35416
compute this resolvent from the coefficients of the polynomial] look very
ugly for order 4 and I know of no feasible approach for higher orders.
--
Ticket URL: <http://trac.sagemath.org/ticket/17516#comment:11>
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