A construction of polynomials with squarefree discriminants

Abstract:

For any integer $n \geq 2$ 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.