2011/03/29 by Kiran S. Kedlaya, Kedlaya, Kiran S. · 3 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Primary 11C08 #Secondary 11R29
paper · pdf · doi:10.48550/arxiv.1103.5728
openalex publication_date 2011/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Sn, 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 ≥ 3, infinitely many quadratic number fields admit everywhere unramified degree n extensions whose normal closures have Galois group An. 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.