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

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).