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.

**1.**Avner Ash, Jos Brakenhoff, and Theodore Zarrabi,*Equality of polynomial and field discriminants*, Experiment. Math.**16**(2007), no. 3, 367–374. MR**2367325****2.**Manjul Bhargava,*The density of discriminants of quartic rings and fields*, Ann. of Math. (2)**162**(2005), no. 2, 1031–1063. MR**2183288**, 10.4007/annals.2005.162.1031**3.**Manjul Bhargava,*The density of discriminants of quintic rings and fields*, Ann. of Math. (2)**172**(2010), no. 3, 1559–1591. MR**2745272**, 10.4007/annals.2010.172.1559**4.**M. Bhargava, The geometric squarefree sieve and unramified nonabelian extensions of quadratic fields, preprint (2011).**5.**Jordan S. Ellenberg and Akshay Venkatesh,*The number of extensions of a number field with fixed degree and bounded discriminant*, Ann. of Math. (2)**163**(2006), no. 2, 723–741. MR**2199231**, 10.4007/annals.2006.163.723**6.**J. Elstrodt, F. Grunewald, and J. Mennicke,*On unramified 𝐴_{𝑚}-extensions of quadratic number fields*, Glasgow Math. J.**27**(1985), 31–37. MR**819826**, 10.1017/S0017089500006054**7.**Andrew Granville,*𝐴𝐵𝐶 allows us to count squarefrees*, Internat. Math. Res. Notices**19**(1998), 991–1009. MR**1654759**, 10.1155/S1073792898000592**8.**George Greaves,*Power-free values of binary forms*, Quart. J. Math. Oxford Ser. (2)**43**(1992), no. 169, 45–65. MR**1150469**, 10.1093/qmath/43.1.45**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), no. 4, 349–402. MR**2099831**, 10.4064/aa115-4-3**11.**C. Hooley,*On the power free values of polynomials*, Mathematika**14**(1967), 21–26. MR**0214556****12.**Takeshi Kondo,*Algebraic number fields with the discriminant equal to that of a quadratic number field*, J. Math. Soc. Japan**47**(1995), no. 1, 31–36. MR**1304187**, 10.2969/jmsj/04710031**13.**Jin Nakagawa,*Binary forms and orders of algebraic number fields*, Invent. Math.**97**(1989), no. 2, 219–235. MR**1001839**, 10.1007/BF01389040**14.**Jin Nakagawa,*Binary forms and unramified 𝐴_{𝑛}-extensions of quadratic fields*, J. Reine Angew. Math.**406**(1990), 167–178. MR**1048239**, 10.1515/crll.1990.406.167

Jin Nakagawa,*Correction to the paper: “Binary forms and unramified 𝐴_{𝑛}-extensions of quadratic fields” [J. Reine Angew. Math. 406 (1990), 167–178; MR1048239 (91d:11037)]*, J. Reine Angew. Math.**413**(1991), 220. MR**1089804****15.**Bjorn Poonen,*Squarefree values of multivariable polynomials*, Duke Math. J.**118**(2003), no. 2, 353–373. MR**1980998**, 10.1215/S0012-7094-03-11826-8**16.**Kôji Uchida,*Unramified extensions of quadratic number fields. II*, Tôhoku Math. J. (2)**22**(1970), 220–224. MR**0272760****17.**Yoshihiko Yamamoto,*On unramified Galois extensions of quadratic number fields*, Osaka J. Math.**7**(1970), 57–76. MR**0266898****18.**Ken Yamamura,*On unramified Galois extensions of real quadratic number fields*, Osaka J. Math.**23**(1986), no. 2, 471–478. MR**856901**

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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:
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