Solvable number field extensions of bounded root discriminant

Leshin, Jonah

doi:10.1090/S0002-9939-2013-12015-0

Solvable number field extensions of bounded root discriminant
HTML articles powered by AMS MathViewer

by Jonah Leshin PDF

Proc. Amer. Math. Soc. 141 (2013), 3341-3352 Request permission

Abstract:

Let $K$ be a number field and $d_K$ the absolute value of the discriminant of $K/\mathbb {Q}$. We consider the root discriminant $d_L^{\frac {1}{[L:\mathbb {Q}]}}$ of extensions $L/K$. We show that for any $N>0$ and any positive integer $n$, the set of length $n$ solvable extensions of $K$ with root discriminant less than $N$ is finite. The result is motivated by the study of class field towers.

References

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, DOI 10.4007/annals.2006.163.723
Farshid Hajir and Christian Maire, Tamely ramified towers and discriminant bounds for number fields. II, J. Symbolic Comput. 33 (2002), no. 4, 415–423. MR 1890578, DOI 10.1006/jsco.2001.0514
David Hilbert, The theory of algebraic number fields, Springer-Verlag, Berlin, 1998. Translated from the German and with a preface by Iain T. Adamson; With an introduction by Franz Lemmermeyer and Norbert Schappacher. MR 1646901, DOI 10.1007/978-3-662-03545-0
Kiran S. Kedlaya, A construction of polynomials with squarefree discriminants, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3025–3033. MR 2917075, DOI 10.1090/S0002-9939-2012-11231-6
Christian Maire, On infinite unramified extensions, Pacific J. Math. 192 (2000), no. 1, 135–142. MR 1741025, DOI 10.2140/pjm.2000.192.135
Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sém. Théor. Nombres Bordeaux (2) 2 (1990), no. 1, 119–141 (English, with French summary). MR 1061762, DOI 10.5802/jtnb.22
Peter Roquette, On class field towers, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 231–249. MR 0218331
J.-P. Serre, Local class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 128–161. MR 0220701
Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237, DOI 10.1007/978-1-4757-5673-9
Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
D. A. Suprunenko, Matrix groups, Translations of Mathematical Monographs, Vol. 45, American Mathematical Society, Providence, R.I., 1976. Translated from the Russian; Translation edited by K. A. Hirsch. MR 0390025, DOI 10.1090/mmono/045
Yoshihiko Yamamoto, On unramified Galois extensions of quadratic number fields, Osaka Math. J. 7 (1970), 57–76. MR 266898