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Solving polynomials by radicals with roots of unity in minimum depth. (English) Zbl 1042.11073

Summary: Let \(k\) be an algebraic number field. Let \(\alpha\) be a root of a polynomial \(f\in k[x]\) which is solvable by radicals. Let \(L\) be the splitting field of \(\alpha\) over \(k\). Let \(n\) be a natural number divisible by the discriminant of the maximal abelian subextension of \(L\), as well as the exponent of \(G(L/k)\), the Galois group of \(L\) over \(k\). We show that an optimal nested radical with roots of unity for \(\alpha\) can be effectively constructed from the derived series of the solvable Galois group of \(L(\zeta_n)\) over \(k(\zeta_n)\).

MSC:

11R32 Galois theory
11Y16 Number-theoretic algorithms; complexity
12Y05 Computational aspects of field theory and polynomials (MSC2010)
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