On Galois Groups of Prime Degree Polynomials with Complex Roots

Abstract

Let $f$ be an irreducible polynomial of prime degree $p\ge 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens Höchsmann (1999), we show that if $p\ge 4k+1$ 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.