Galois extension

Natural density

Linguistics

Philosophy

Discrete mathematics

Algebraic number field

Exponent

Degree (music)

Square-free integer

Computer science

Acoustics

Integer (computer science)

Generic polynomial

Polynomial

Physics

Galois cohomology

Monic polynomial

Mathematics

Discriminant

Mathematical analysis

Artificial intelligence

Combinatorics

Conjecture

Programming language

Galois group

Differential Galois theory

Inventiones mathematicae

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.

Squarefree values of polynomial discriminants I