Squarefree values of polynomial discriminants I

TL;DR

This paper studies the density of monic integer polynomials with squarefree discriminants and irreducibility, establishing positive lower densities and providing explicit formulas, with implications for the distribution of monogenic number fields.

Contribution

It proves the positive lower density of polynomials with squarefree discriminant and irreducibility, and derives bounds on the number of monogenic fields with Galois group S_n.

Findings

Positive lower density of polynomials with squarefree discriminant

Explicit density given by (2)^{-1} for irreducible polynomials

Lower bound on the number of monogenic fields with Galois group S_n

Abstract

We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials $f(x)$, such that $f(x)$ is irreducible and $\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field, is positive, and is in fact given by $\zeta(2)^{-1}$. It also follows from our methods that there are $\gg X^{1/2+1/n}$ monogenic number fields of degree $n$ having associated Galois group $S_n$ and absolute discriminant less than $X$, and we conjecture that the exponent in this lower bound is optimal.