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
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Acknowledgments
The authors thank Dani Neftin for remarks on characteristic polynomials of random matrices and for bringing relevant references to our attention. They thank the reviewer for helpful comments. Some of the work of the second author was done at the University of Michigan and at the Technion, whom he thanks for support.
The first author was partially supported by NSF grants DMS-1101373 and DMS-1401224.
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Lagarias, J.C., Weiss, B.L. Splitting behavior of S n -polynomials. Res. number theory 1, 7 (2015). https://doi.org/10.1007/s40993-015-0006-6
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DOI: https://doi.org/10.1007/s40993-015-0006-6