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

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 $\mathrm{A}\mathrm{f}\mathrm{f}({\mathbb{F}}_p)$ or $C_p⋊C_{(p-1)\mathrm{/}2}⊲\mathrm{A}\mathrm{f}\mathrm{f}({\mathbb{F}}_p)$, and the Galois group of $h$, which is $C_2\times \mathrm{A}\mathrm{f}\mathrm{f}({\mathbb{F}}_p),\mathrm{A}\mathrm{f}\mathrm{f}({\mathbb{F}}_p)$, or $C_2\times (C_p⋊C_{(p-1)\mathrm{/}2})⊲C_2\times \mathrm{A}\mathrm{f}\mathrm{f}({\mathbb{F}}_p)$.