Splitting Behavior of $S_n$-Polynomials

TL;DR

This paper investigates the probability distribution of splitting behaviors of random degree n polynomials with integer coefficients, focusing on Galois groups, discriminants, and prescribed splitting types at a finite set of primes, using splitting measures.

Contribution

It introduces splitting measures to describe the asymptotic probabilities of various splitting behaviors and establishes existence results for degree n extensions with specified local properties.

Findings

Asymptotic probabilities are described by splitting measures evaluated at primes.

Existence of degree n extensions with prescribed local splitting conditions.

Comparison with Bhargava's distributions for prime splitting probabilities.

Abstract

We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial f(x) with integer coefficients in a box of side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with splitting field over the rationals having Galois group $S_n$; (ii) the polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type at each prime p in S. The limit probabilities as $B \to \infty$ are described in terms of values of a one-parameter family of measures on $S_n$, called 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 the rationals 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 $p$.