Lifting monogenic cubic fields to monogenic sextic fields

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}.