Kodai Mathematical Journal

Lifting monogenic cubic fields to monogenic sextic fields

Melisa J. Lavallee, Blair K. Spearman, and Kenneth S. Williams

Full-text: Open access

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}.

Article information

Source
Kodai Math. J., Volume 34, Number 3 (2011), 410-425.

Dates
First available in Project Euclid: 10 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1320935550

Digital Object Identifier
doi:10.2996/kmj/1320935550

Mathematical Reviews number (MathSciNet)
MR2855831

Zentralblatt MATH identifier
1237.11045

Citation

Lavallee, Melisa J.; Spearman, Blair K.; Williams, Kenneth S. Lifting monogenic cubic fields to monogenic sextic fields. Kodai Math. J. 34 (2011), no. 3, 410--425. doi:10.2996/kmj/1320935550. https://projecteuclid.org/euclid.kmj/1320935550


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