Squarefree values of polynomial discriminants I

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.