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Linear forms in the logarithms of algebraic numbers (III)

  • A. Baker (a1)

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.

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page 220 note † Baker, A., “Linear forms in the logarithms of algebraic numbers”, Mathematika, 13 (1966), 204216, (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.

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

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

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.

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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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