The North Carolina Journal of Mathematics and Statistics

Subfields of Solvable Sextic Field Extensions

Chad Awtrey, Peter Jakes

Abstract


Let F be a field, f(x) in F[x] an irreducible polynomial of degree six, K the stem field of f, and G the Galois group of f over F. We show G is solvable if and only if K/F has either a quadratic or cubic subfield. We also show that G can be determined by: the size of the automorphism group of K/F, the discriminant of f, and the discriminants of polynomials defining intermediate fields. Since most methods for computing polynomials defining intermediate subfields require factoring f over its stem, we include a method that does not require factorization over K, but rather only relies factoring two linear resolvent polynomials over F.

Keywords


sextic polynomials; Galois group computation; subfields; linear resolvents

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