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 and any nonnegative integers with , we give an unconditional construction of infinitely many monic irreducible polynomials of degree with integer coefficients having squarefree discriminant and exactly real roots. These give rise to number fields of degree , signature , Galois group , 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 is at least . A corollary is that for each , infinitely many quadratic number fields admit everywhere unramified degree extensions whose normal closures have Galois group . This generalizes results of Yamamura, who treats the case , and Uchida and Yamamoto, who allow general but do not control the real place.

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Additional Information

**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

Email:
kedlaya@mit.edu, kedlaya@ucsd.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11231-6

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