On Galois Groups of Prime Degree Polynomials with Complex Roots

TL;DR

This paper establishes conditions under which the Galois group of prime degree polynomials with complex roots is either symmetric or alternating, improving algorithms for Galois group computation and characterizing solvable cases.

Contribution

It improves the algorithm for computing Galois groups of prime degree polynomials and characterizes when these groups are Frobenius groups based on complex roots.

Findings

If p ≥ 4k+1, Galois group is A_p or S_p.

Solvable polynomials have Frobenius Galois groups.

Any even Frobenius group of degree p can be realized as a Galois group.

Abstract

Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens H\"{o}chsmann (1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ is isomorphic to $A_{p}$ or $S_{p}$. This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T.Shaska. If such a polynomial $f$ is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of an irreducible polynomial of degree $p$ over $\QQ$ having complex roots.