A construction of polynomials with squarefree discriminants

TL;DR

This paper presents an explicit method to construct infinitely many degree n polynomials with integer coefficients, squarefree discriminant, and specified real roots, leading to new insights into number fields and their Galois groups.

Contribution

It provides an unconditional construction of polynomials with squarefree discriminants and prescribed real roots, extending previous results to all n ≥ 3.

Findings

Constructs infinitely many such polynomials for each n ≥ 2

Produces number fields with Galois group S_n and squarefree discriminant

Shows existence of infinitely many quadratic fields with specific unramified extensions

Abstract

For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r real roots. These give rise to number fields of degree n, signature (r,s), Galois group S_n, and squarefree discriminant; we may also force the discriminant to be coprime to any given integer. The number of fields produced with discriminant in the range [-N, N] is at least c N^(1/(n-1)). A corollary is that for each n \geq 3, infinitely many quadratic number fields admit everywhere unramified degree n extensions whose normal closures have Galois group A_n. This generalizes results of Yamamura, who treats the case n = 5, and Uchida and Yamamoto, who allow general n but do not control the real place.