Probabilistic Galois theory over p-adic fields

We estimate several probability distributions arising from the study of random, monic polynomials of degree n with coefficients in the integers of a general p-adic field K-p having residue field with q = p(f) elements. We estimate the distribution of the degrees of irreducible factors of the polynomials, with tight error bounds valid when q > n(2) + n. We also estimate the distribution of Galois groups of such polynomials, showing that for fixed n, almost all Galois groups are cyclic in the limit q -> infinity. In particular, we show that the Galois groups are cyclic with probability at least 1 - 1/q. We obtain exact formulas in the case of K-p for all p > n when n = 2 and n = 3. (C) 2012 Elsevier Inc. All rights reserved.