A construction of polynomials with squarefree discriminants
Author:
Kiran S. Kedlaya
Journal:
Proc. Amer. Math. Soc. 140 (2012), 30253033
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/S000299392012112316
Received by editor(s):
March 29, 2011
Published electronically:
January 23, 2012
Additional Notes:
The author was supported by NSF CAREER grant DMS0545904, DARPA grant HR00110910048, 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
