GALOIS GROUP OF Xpn+aX+a

Abstract

Let $p$ be an odd prime number, $n\ge 3$ be a rational integer, and let $f(X)=X^{p^n}+aX+a\in \mathbb{Z}[X]$ be an Eisenstein trinomial with respect to $p$. We prove that the Galois group $G$ of $f(X)$ over the field $\mathbb{Q}$ of rational numbers, is either the full symmetric group $S_{p^n}$, or $AGL(1,p^n)\le G\le AGL(n,p)$. We also show that $G\simeq S_{p^n}$, except possibly when $|p^{n{p}^n-1}+\frac{a}{p}{(p^n-1)}^{p^n-1}|$ is a square, and for each prime divisor $\ell$ of $a∕p$, $p$ divides the $\ell$-adic valuation $v_{\ell }(a)$ of the integer $a$.