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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 $C_2 \times (C_p \rtimes C_{(p - 1)/2}) \vartriangleleft C_2 \times\mathrm{Aff}(\mathbb{F}_p)$.

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