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

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

Published electronically:
January 23, 2012

MathSciNet review:
2917075

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**A. Ash, J. Brakenhoff, and T. Zarrabi, Equality of polynomial and field discriminants,*Exper. Math.***16**(2007), 367-374. MR**2367325 (2008i:11129)****2.**M. Bhargava, The density of discriminants of quartic rings and fields,*Ann. of Math. (2)***162**(2005), 1031-1063. MR**2183288 (2006m:11163)****3.**M. Bhargava, The density of discriminants of quintic rings and fields,*Ann. of Math. (2)***172**(2010), 1559-1591. MR**2745272****4.**M. Bhargava, The geometric squarefree sieve and unramified nonabelian extensions of quadratic fields, preprint (2011).**5.**J.S. Ellenberg and A. Venkatesh, The number of extensions of a number field with fixed degree and bounded discriminant,*Ann. of Math. (2)***163**(2006), 723-741. MR**2199231 (2006j:11159)****6.**J. Elstrodt, F. Grunewald, and J. Mennicke, On unramified -extensions of quadratic number fields,*Glasgow Math. J.***27**(1985), 31-37. MR**819826 (87e:11122)****7.**A. Granville, allows us to count squarefrees,*Intl. Math. Res. Notices***1998**, 991-1009. MR**1654759 (99j:11104)****8.**G. Greaves, Power-free values of binary forms,*Quart. J. Math. Oxford***43**(1992), 45-65. MR**1150469 (92m:11098)****9.**G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, fourth edition, Oxford University Press (London), 1965.**10.**H.A. Helfgott, On the square-free sieve,*Acta Arith.***115**(2004), 349-402. MR**2099831 (2005h:11211)****11.**C. Hooley, On the power free values of polynomials,*Mathematika***14**(1967), 21-26. MR**0214556 (35:5405)****12.**T. Kondo, Algebraic number fields with the discriminant equal to that of a quadratic number field,*J. Math. Soc. Japan***47**(1995), 31-36. MR**1304187 (95h:11121)****13.**J. Nakagawa, Binary forms and orders of algebraic number fields,*Invent. Math.***97**(1989), 219-235; erratum,*Invent. Math.***105**(1991), 443. MR**1001839 (90k:11042)****14.**J. Nakagawa, Binary forms and unramified -extensions of quadratic fields,*J. Reine Angew. Math.***406**(1990), 167-178; correction,*J. Reine Angew. Math.***413**(1991), 220. MR**1048239 (91d:11037)**; MR**1089804 (91j:11019)****15.**B. Poonen, Squarefree values of multivariate polynomials,*Duke Math. J.***118**(2003), 353-373. MR**1980998 (2004d:11094)****16.**K. Uchida, Unramified extensions of quadratic number fields, II,*Tôhoku Math. J.***22**(1970), 220-224. MR**0272760 (42:7641)****17.**Y. Yamamoto, On unramified Galois extensions of quadratic number fields,*Osaka J. Math.***7**(1970), 57-76. MR**0266898 (42:1800)****18.**K. Yamamura, On unramified Galois extensions of real quadratic number fields,*Osaka J. Math.***23**(1986), 471-478. MR**856901 (88a:11112)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
11C08,
11R29

Retrieve articles in all journals with MSC (2010): 11C08, 11R29

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:
https://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