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

Abstract

We prove an irreducibility criterion for polynomials of the form $h(x)=x^{2m}+b{x}^m+c_1\in F[x]$ 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 $C_p⋊C_{(p-1)/2}⊲\mathrm{Aff}({\mathbb{F}}_p)$, and the Galois group of $h$, which is $C_2\times \mathrm{Aff}({\mathbb{F}}_p),\mathrm{Aff}({\mathbb{F}}_p)$, or $C_2\times (C_p⋊C_{(p-1)/2})⊲C_2\times \mathrm{Aff}({\mathbb{F}}_p)$.

15 pages, 3 figures