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 Kp having residue field with q=pf elements. We estimate the distribution of the degrees of irreducible factors of the polynomials, with tight error bounds valid when q>n2+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 1−1q. We obtain exact formulas in the case of Kp for all p>n when n=2 and n=3.