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

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 groups of $g_4(x)$ and $g_6(x)$, respectively. This paper provides a streamlined elementary characterization of all sixteen possible Galois groups of $f(x)$. In particular, we give explicit criteria showing 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. This contributes to a broader program to systematically study the Galois groups of $x^{2m}+ax^m+b$ for a positive integer $m$.