Galois groups of prime degree polynomials with nonreal roots

Abstract

In the process of computing the Galois group of a prime degree polynomial $f(x)$ over $\mathbb{Q}$ we suggest a preliminary checking for the existence of non-real roots. If $f(x)$ has non-real roots, then combining a 1871 result of Jordan and the classification of transitive groups of prime degree which follows from CFSG we get that the Galois group of $f(x)$ contains $A_p$ or is one of a short list. Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of prime degree $p\ge 5$ and $r=2s$ be the number of non-real roots of $f(x)$. We show that if $s$ satisfies $s(s\log s+2\log s+3)\le p$ then $Gal(f)=A_p,S_p$.