Splitting behavior of S n -polynomials

• Published in: Research in number theory Vol. 1; no. 1
• Main Authors: Lagarias, Jeffrey C, Weiss, Benjamin L
• Format: Journal Article
• Language: English
• Published: 01.12.2015
• ISSN: 2363-9555; 2363-9555
• DOI: 10.1007/s40993-015-0006-6

We analyze the probability that, for a fixed finite set of primes S , a random, monic, degree n polynomial $f(x) \in {\mathbb {Z}}[x]$ f ( x ) ∈ ℤ [ x ] with coefficients in a box of side B satisfies: (i) f ( x ) is irreducible over "Image missing" , with splitting field $K_{f}/{\mathbb {Q}}$ K f / Q over "Image missing" 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 "Image missing" 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