On Splitting Types, Discriminant Bounds, and Conclusive Tests for the Galois Group

TL;DR

This paper generalizes fundamental algebraic number theory results related to Galois groups, providing new bounds and conclusive tests for determining Galois groups of polynomials with integer coefficients.

Contribution

It removes artificial constraints from the Kummer-Dedekind Theorem and establishes an elementary, computable upper bound for discriminants, enabling finite-step determination of Galois groups.

Findings

Removal of constraints from the Kummer-Dedekind Theorem

Elementary proof of discriminant bounds in terms of polynomial coefficients

Finite-step determination of Galois groups for cubics, quartics, and quintics

Abstract

Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types of polynomials modulo primes, and cycle types of the Galois groups of polynomials. One remarkable example is the removal of all artificial constraints from the Kummer-Dedekind Theorem that relates splitting and factorization patterns. Finally, we present an elementary proof that the discriminant of the splitting field of a monic irreducible polynomial with integer coefficients has a computable upper bound in terms of the coefficients. This result, combined with one of Lagarias et al., shows that tests of polynomials for the cycle types of the Galois group are conclusive. In particular, the Galois groups of monic irreducible cubics, quartics, and quintics with integer coefficients can be completely determined in finitely many steps (though not necessarily in one's lifetime).