On Galois Groups of Prime Degree Polynomials with Complex Roots
Oz Ben-Shimol

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 be an irreducible polynomial of prime degree over , with precisely pairs of complex roots. Using a result of Jens H\"{o}chsmann (1999), we show that if then is isomorphic to or . 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 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 over having complex roots.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Cryptography and Residue Arithmetic · Polynomial and algebraic computation
