Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-05T08:35:23.898Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear forms in the logarithms of algebraic numbers

Published online by Cambridge University Press:  26 February 2010

A. Baker
A. Baker
Affiliation:
Trinity College, Cambridge
Get access

Extract

In 1934 Gelfond [2] and Schneider [6] proved, independently, that the logarithm of an algebraic number to an algebraic base, other than 0 or 1, is either rational or transcendental and thereby solved the famous seventh problem of Hilbert. Among the many subsequent developments (cf. [4, 7, 8]), Gelfond [3] obtained, by means of a refinement of the method of proof, a positive lower bound for the absolute value of β1 log α1+β2 log α2, where β1, β2 denote algebraic numbers, not both 0, and α1,α1, denote algebraic numbers not 0 or 1, with log α1/log α2 irrational. Of particular interest is the special case in which β1,β2 denote integers. In this case it is easy to obtain a trivial positive lower bound (cf. [1; Lemma 2]), and the existence of a non-trivial bound follows from the Thue–Siegel–Roth theorem (see [4; Ch. I]). But Gelfond's result improves substantially on the former, and, unlike the latter, it is derived by an effective method of proof. Gelfond [4; p. 177] remarked that an analogous theorem for linear forms in arbitrarily many logarithms of algebraic numbers would be of great value for the solution of some apparently very difficult problems of number theory. It is the object of this paper to establish such a result.

Type
Research Article
Information
Mathematika , Volume 13 , Issue 2 , December 1966 , pp. 204 - 216
Copyright
Copyright © University College London 1966

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. Feldman, N. I., “Approximation by algebraic numbers to the logarithms of algebraic numbers”, Izv. Akad. Nauk SSSR, 24 (1960), 475–492 (in Bussian).Google Scholar
2. Gelfond, A. O., “On Hilbert's seventh problem”, Doklady Akad. Nauk SSSR, 2 (1934), 1–6;Google Scholar
Izv. Akad. Nauk SSSR, 7 (1934), 623–634 (in Russian and French).Google Scholar
3. Gelfond, A. O., “On the approximation of the ratio of logarithms of two algebraic numbers by algebraic numbers”, Izv. Akad. Nauk SSSR, 5–6 (1939), 509–518 (in Russian).Google Scholar
4. Gelfond, A. O., Transcendental and algebraic numbers (New York, 1960).Google Scholar
5. Gelfond, A. O. and Linnik, Yu. V. “On Thue's method and the problem of effectiveness in quadratic fields”, Doklady Akad. Nauk SSSR, 61 (1948), 773–776 (in Russian).Google Scholar
6. Schneider, Th., “Transzendenzuntersuchungen periodischer Funktionen I. Transzendenz von Potenzen”, J. Reine Angew. Math. 172 (1934), 65–69.Google Scholar
7. Schneider, Th., Einführung in die transzendenten Zahlen (Berlin, Göttingen, Heidelberg, 1957).Google Scholar
8. Siegel, C. L., Transcendental numbers (Princeton, 1949).Google Scholar