Open Access
October 2011 Lifting monogenic cubic fields to monogenic sextic fields
Melisa J. Lavallee, Blair K. Spearman, Kenneth S. Williams
Kodai Math. J. 34(3): 410-425 (October 2011). DOI: 10.2996/kmj/1320935550

Abstract

Let e $\in$ {-1, +1}. Let a,b $\in$ Z be such that x6 + ax4 + bx2 + e is irreducible in Z[x]. The cubic field C = Q (α), where α3 + aα2 + bα + e = 0, is said to lift to the sextic field K = Q(θ), where θ6 + aθ4 + bθ2 + e = 0. The field K is called the lift of C. If {1, α, α2} is an integral basis for C (so that C is monogenic), we investigate conditions on a and b so that {1, θ, θ2, θ3, θ4, θ5} is an integral basis for the lift K of C (so that K is monogenic). As the sextic field K contains a cubic subfield (namely C), there are eight possibilities for the Galois group of K. For five of these Galois groups, we show that infinitely many monogenic sextic fields can be obtained in this way, and for the remaining three Galois groups, we show that only finitely many monogenic fields can arise in this way, when e $\in$ {-1, +1}.

Citation

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Melisa J. Lavallee. Blair K. Spearman. Kenneth S. Williams. "Lifting monogenic cubic fields to monogenic sextic fields." Kodai Math. J. 34 (3) 410 - 425, October 2011. https://doi.org/10.2996/kmj/1320935550

Information

Published: October 2011
First available in Project Euclid: 10 November 2011

zbMATH: 1237.11045
MathSciNet: MR2855831
Digital Object Identifier: 10.2996/kmj/1320935550

Rights: Copyright © 2011 Tokyo Institute of Technology, Department of Mathematics

Vol.34 • No. 3 • October 2011
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