# Splitting behavior of *S*
_{
n
}-polynomials

- Jeffrey C Lagarias
^{1}Email author and - Benjamin L Weiss
^{2}

**1**:7

**DOI: **10.1007/s40993-015-0006-6

© Legarias and Weiss.; licensee Springer. 2015

**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