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Proceedings of the American Mathematical Society

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A construction of polynomials with squarefree discriminants

Author: Kiran S. Kedlaya
Journal: Proc. Amer. Math. Soc. 140 (2012), 3025-3033
MSC (2010): Primary 11C08; Secondary 11R29
Published electronically: January 23, 2012
MathSciNet review: 2917075
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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.

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Kiran S. Kedlaya
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 – and – Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093

Received by editor(s): March 29, 2011
Published electronically: January 23, 2012
Additional Notes: The author was supported by NSF CAREER grant DMS-0545904, DARPA grant HR0011-09-1-0048, MIT (NEC Fund, Green Career Development Professorship), and UC San Diego (Warschawski Professorship).
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 Kiran S. Kedlaya