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Algebraic independence of some values of the exponential function

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We prove general results concerning the algebraic independence of three values of the exponential function. Forβ algebraic and of degree 7 andα algebraic and ≠ 0, 1 there exist among the numbers αβ,...,\(\alpha ^{\beta ^6 } \) three which are algebraically independent. The proof employs a method due to A. O. Gel'fond and N. I. Fel'dman.

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Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 661–672, April, 1974.

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Chudnovskii, G.V. Algebraic independence of some values of the exponential function. Mathematical Notes of the Academy of Sciences of the USSR 15, 391–398 (1974).

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  • General Result
  • Exponential Function
  • Algebraic Independence