Open Access

Splitting behavior of S n -polynomials

Research in Number Theory20151:7

https://doi.org/10.1007/s40993-015-0006-6

Received: 13 January 2015

Accepted: 24 January 2015

Published: 9 July 2015

Abstract

We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial \(f(x) \in {\mathbb {Z}}[x]\) with coefficients in a box of side B satisfies: (i) f(x) is irreducible over , with splitting field \(K_{f}/{\mathbb {Q}}\) over having Galois group S n ; (ii) the polynomial discriminant D i s c(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type (mod p) at each prime p in S.

The limit probabilities as B are described in terms of values of a one-parameter family of measures on S n , called z-splitting measures, with parameter z evaluated at the primes p in S. We study properties of these measures.

We deduce that there exist degree n extensions of with Galois closure having Galois group S n with a given finite set of primes S having given Artin symbols, with some restrictions on allowed Artin symbols for p<n. We compare the distributions of these measures with distributions formulated by Bhargava for splitting probabilities for a fixed prime p in such degree n extensions ordered by size of discriminant, conditioned to be relatively prime to p.

Mathematics Subject Classification:Primary 11R09; Secondary 11R32; 12E20; 12E25