A construction of polynomials with squarefree discriminants

For any integer n ≥ 2 n ≥ 2 and any nonnegative integers r , s r,s with r + 2 s = n r+2s = n , we give an unconditional construction of infinitely many monic irreducible polynomials of degree n n with integer coefficients having squarefree discriminant and exactly r r real roots. These give rise to number fields of degree n n , signature ( r , s ) (r,s) , Galois group S n 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 ] [-N, N] is at least c N 1 / ( n − 1 ) c N1/(n-1) . A corollary is that for each n ≥ 3 n ≥ 3 , infinitely many quadratic number fields admit everywhere unramified degree n n extensions whose normal closures have Galois group A n A_n . This generalizes results of Yamamura, who treats the case