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Linear forms in the logarithms of algebraic numbers (III)

Published online by Cambridge University Press:  26 February 2010

A. Baker
A. Baker
Affiliation:
Trinity College, Cambridge.
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Extract

In the present paper the researches initiated in the two earlier papers of this series are continued, and, by suitable generalizations of the techniques employed therein, solutions are obtained to some further well known problems from the theory of transcendental numbers. It will be proved, for example, that a non-vanishing linear form, with algebraic coefficients, in the logarithms of algebraic numbers, cannot be algebraic. This implies, in particular, that π + log α is transcendental for any algebraic number α ≠ 0, and also eα π+ß is transcendental for all algebraic numbers α, β with β ≠ 0.

Type
Research Article
Information
Mathematika , Volume 14 , Issue 2 , December 1967 , pp. 220 - 228
Copyright
Copyright © University College London 1967

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References

page 220 note † Baker, A., “Linear forms in the logarithms of algebraic numbers”, Mathematika, 13 (1966), 204–216CrossRefGoogle Scholar, (II) 14 (1967), 102–107. The papers will be referred to as (I) and (II) respectively.

page 220 note ‡ The height of an algebraic number is given, as in (I), by the maximum of the absolute values of the relatively prime integer coefficients in the minimal defining polynomial.

page 220 note § For log α1, …, log αn we take any fixed determinations of the logarithms. The value of C depends on the choice of these determinations.

page 221 note † Siegel, C. L., Transcendental numbers (Princeton, 1949); see pp. 84 and 97.Google Scholar

page 221 note ‡ See Gelfond, A. O., Transcendental and algebraic numbers (New York, 1960), Theorem III, p. 134.Google Scholar

page 221 note § See Baker, A., “On the representation of integers by binary forms” to appear in Phil. Trans. Royal Society London, series A.Google Scholar

page 225 note † cf. Gelfond; loc. cit. pp. 140-142.

page 226 note † Here, as in the next displayed formula, i denotes (− 1)½ otherwise i represents a variable suffix.