Gelfond–Schneider theorem

History

Statement

Comments

• The values of a and b are not restricted to real numbers; complex numbers are allowed.
• In general, is multivalued, where log stands for the complex logarithm. This accounts for the phrase "any value of" in the theorem's statement.
• An equivalent formulation of the theorem is the following: if α and γ are nonzero algebraic numbers, and we take any non-zero logarithm of α, then is either rational or transcendental. This may be expressed as saying that if, are linearly independent over the rationals, then they are linearly independent over the algebraic numbers. The generalisation of this statement to more general linear forms in logarithms of several algebraic numbers is in the domain of transcendental number theory.
• If the restriction that a and b be algebraic is removed, the statement does not remain true in general. For example,
• Kurt Mahler proved the p-adic analogue of the theorem: if a and b are in C_p, the completion of the algebraic closure of Q_p, and they are algebraic over Q, and if and then is either rational or transcendental, where log_p is the p-adic logarithm function.

  Corollaries

• Gelfond–Schneider constant and its square root
• Gelfond's constant

• Applications