INTERSECTIVE POLYNOMIALS WITH GALOIS GROUP D₅

We give an infinite family of intersective polynomials with Galois group D5, the dihedral group of order 10

Lifting monogenic cubic fields to monogenic sextic fields

Let e ∈ {-1, +1}. Let a,b ∈ 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 ∈ {-1, +1}