A construction of polynomials with squarefree discriminants
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- by Kiran S. Kedlaya PDF
- Proc. Amer. Math. Soc. 140 (2012), 3025-3033
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.References
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Additional Information
- Kiran S. Kedlaya
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 โ and โ Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093
- MR Author ID: 349028
- ORCID: 0000-0001-8700-8758
- Email: kedlaya@mit.edu, kedlaya@ucsd.edu
- 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
- © Copyright 2012 Kiran S. Kedlaya
- Journal: Proc. Amer. Math. Soc. 140 (2012), 3025-3033
- MSC (2010): Primary 11C08; Secondary 11R29
- DOI: https://doi.org/10.1090/S0002-9939-2012-11231-6
- MathSciNet review: 2917075