A construction of polynomials with squarefree discriminants

Abstract

For any integer n ≥ 2 and any nonnegative integers r, swith 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[subscript 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 cN[superscript 1/(n-1)]. A corollary is that for each n ≥ 3, infinitely many quadratic number fields admit everywhere unramified degree n extensions whose normal closures have Galois group A[subscript 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.