A construction of polynomials with squarefree discriminants

Details A construction of polynomials with squarefree discriminants A construction of polynomials with squarefree discriminants

2011-03-29

arxiv.org/abs/1103.5728v2

Mathematics

Number Theory

10 pages; v2: refereed version, minor edits only

Scientific paper

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.

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