On the Galois group of a reciprocal even octic polynomial

Abstract

Let F be a field of characteristic 0, and let $f(x)=x^8+a{x}^6+b{x}^4+a{x}^2+1\in F[x]$ be irreducible with Galois group $\text{Gal}(f)$. We extend results in the literature to give a complete and elementary classification of $\text{Gal}(f)$, identified as a transitive subgroup of S₈ up to conjugacy. The approach involves testing whether a small number of elements in F are perfect squares. As an application, we give one-parameter families of polynomials for each of the six possible Galois groups.

KEYWORDS:

• Computation
• discriminant
• Galois group
• octic polynomials

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 11Y40
• 12F10