The Galois Group of $x^{2p}+bx^p+c^p$ over $\mathbb{Q}$

Abstract

We prove an irreducibility criterion for polynomials of the form $h(x)=x^{2m} + bx^m + c_1 \in F[x]$ relating to the Dickson polynomials of the first kind $D_p$. In the case when $F = \mathbb{Q}$, $m$ is a prime $p>3$, and $c_1=c^p$, for $c\in\mathbb{Q}$, we explicitly determine the Galois group of $d_h= D_p(x, c) + b$, which is $\mathrm{Aff}(\mathbb{F}_p)$ or $C_p \rtimes C_{(p - 1)/2} \vartriangleleft \mathrm{Aff}(\mathbb{F}_p)$, and the Galois group of $h$, which is $C_2 \times \mathrm{Aff}(\mathbb{F}_p), \mathrm{Aff}(\mathbb{F}_p)$, or $C_2 \times (C_p \rtimes C_{(p - 1)/2}) \vartriangleleft C_2 \times \mathrm{Aff}(\mathbb{F}_p)$.