On Galois groups of prime degree polynomials with complex roots

Abstract

Let \(f\) be an irreducible polynomial of prime degree \(p\geq 5\) over \({\mathbb Q}\), with precisely \(k\)  pairs of complex roots. Using a result of Jens Hochsmann (1999), show that if  \(p\geq 4k+1\) then \(\operatorname{Gal}(f/{\mathbb Q})\) 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 \({\mathbb Q}\) having complex roots.