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 $K_{\mathfrak{p}}$ 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 $q\to \infty$. In particular, we show that the Galois groups are cyclic with probability at least $1-\frac{1}{q}$. We obtain exact formulas in the case of $K_{\mathfrak{p}}$ for all $p>n$ when $n=2$ and $n=3$.

27 pages