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

Figures (3)

• Figure 1: The field diagram when $\sqrt{\Delta} \notin F_p$
• Figure 2: The field diagram when $\sqrt{\Delta} \in F_p$ and $p \equiv 1 \bmod 4$
• Figure 3: The field diagram when $\sqrt{\Delta} \in F_p$ and $p \equiv 3 \bmod 4$.

Theorems & Definitions (43)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Example 1
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Theorem 2.1: Capelli's Theorem
• Theorem 2.2: Capelli