Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T06:47:41.576Z Has data issue: false hasContentIssue false
  • English
  • Français

The number of generators of the integers of a number field

Published online by Cambridge University Press:  26 February 2010

P. A. B. Pleasants
P. A. B. Pleasants
Affiliation:
University College, Cardiff.
Get access

Extract

This paper deals with the problem (raised by J. Browkin) of how many ring generators are needed for the ring of integers of a given algebraic number field. I show that the number of generators needed for the integers of a field of degree n is less than (logn/log2) + 1, and that if 2 splits completely in the field the number of generators needed is in fact the largest integer less than (logn/log2) + 1. These results follow from a computable formula (that depends only on how the small primes factorize in the field) for the number of generators of the ring of integers of a given field. This formula has the single drawback that when it yields “one” two generators may be needed, and I show that there are fields of arbitrarily high degree for which this happens.

Type
Research Article
Information
Mathematika , Volume 21 , Issue 2 , December 1974 , pp. 160 - 167
Copyright
Copyright © University College London 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bauer, M.. “Über die ausserwesentlichen Diskriminantenteiler einer Gattung”, Math. Ann., 64 (1907), 573–576.Google Scholar
2.Dickson, L. E., Mitchell, H. H., Vandiver, H. S. and Wahlin, G. E.. “Algebraic numbers”, USA National Research Council Bulletin No. 28 (1923). Reprinted in Algebraic numbers (Chelsea, 1968).Google Scholar
3.FriShlich, A.. “Local fields”, in Algebraic number theory, ed. Cassels, J. W. S. and Fröhlich, A. (Academic Press, 1967).Google Scholar
4.Hall, Marshall. “Indices in cubic fields”, Bull. Amer. Math. Soc., 43 (1937), 104–108.Google Scholar
5.Hasse, H.. Zahlentheorie, third edition (Akademie-Verlag, Berlin, 1969).Google Scholar
6.Hensel, K.. “Arithmetische Untersuchungen über die gemeinsamen ausserwesentlichen Discriminantentheiler einer Gattung”, J. Reine Angew. Math., 113 (1894), 128–160.Google Scholar
7.Hensel, K.. “Über die Fundamentalgleichung und die ausserwesentlichen Discriminantentheiler eines algebraischen Körpers”, Nachr. Ges. Wiss. Göttingen. Math. Phys. Kl.,(1897), 254–260.Google Scholar
8.Mann, H. B.. Introduction to algebraic number theory (Ohio State University, Columbus, Ohio, 1955).Google Scholar
9.Selmer, E. S.. “The Diophantine equation ax 3 + by 3 + cz 3 ≤ 0”, Acta Math., 85 (1951), 203–362.Google Scholar
10.Skolem, T.. “Unlösbarkeit von Gleichungen, deren entsprechende Kongruenz für jeden Modul lösbar ist”, Oslo Vid. Akad. Avh., I, Mat. nat. Kl., (1942) No. 4.Google Scholar
11.von Źyliński, E.. “Zur Theorie der ausserwesentlichen Diskriminantenteiler algebraischer Körper”, Math. Ann., 73 (1913), 273–274.Google Scholar