Elementary characterization for Galois groups of $x^{12}+ax^6+b$

TL;DR

This paper provides an elementary method to determine the Galois group of the polynomial $x^{12}+ax^6+b$ over $\

Contribution

It introduces a new elementary characterization of all possible Galois groups of the polynomial based on known groups of related lower-degree polynomials.

Findings

Galois group of $f(x)$ is determined by $(G_4,G_6)$ and two rational square tests.

Complete classification of 16 possible Galois groups for the polynomial.

Method simplifies Galois group determination for this class of polynomials.

Abstract

Let $f(x)=x^{12}+ax^6+b \in \mathbb{Q}[x]$ be an irreducible polynomial, $g_4(x)=x^4+ax^2+b$, $g_6(x)=x^6+ax^3+b$, and let $G_4$ and $G_6$ be the Galois group of $g_4(x)$ and $g_6(x)$, respectively. Building upon known characterizations of $G_4$ and $G_6$ in the literature, this paper provides an elementary characterization of all sixteen possible Galois groups of $f(x)$. In particular, we show that the Galois group of $f(x)$ can be uniquely determined by the pair $(G_4,G_6)$ along with testing whether at most two expressions involving $a$ and $b$ are rational squares.