Galois groups of certain even octic polynomials

TL;DR

This paper provides a complete classification of Galois groups for a family of even octic polynomials over rationals, using simple arithmetic conditions, and applies these results to specific polynomial forms.

Contribution

It introduces a method to determine factorization patterns and classify all possible Galois groups for certain even octic polynomials, extending to specific polynomial families.

Findings

Six possible Galois groups identified and classified.

Method for analyzing factorization patterns using arithmetic conditions.

Application to polynomials of the form x^8+ax^4+1 and similar.

Abstract

Let $f(x)=x^8+ax^4+b \in \mathbb{Q}[x]$ be an irreducible polynomial where $b$ is a square. We give a method that completely describes the factorization patterns of a linear resolvent of $f(x)$ using simple arithmetic conditions on $a$ and $b$. As a result, we determine the exact six possible Galois groups of $f(x)$ and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials $x^8+ax^4+1 \in \mathbb{Q}[x]$. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials $x^8+ax^6+bx^4+ax^2+1 \in \mathbb{Q}[x]$.