Probabilistic Galois theory over p -adic fields

Abstract

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 → ∞ . 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 .