Theorem of Gelfond and Schneider

Abstract

The twentieth century turned out to be a golden era for transcendental number theory. Several important results were proved during this period. In August 1900, at the second International Congress of Mathematicians (ICM) held in Paris, Hilbert proposed a list of 23 problems which, according to him, will have profound value for the progress of mathematical sciences. The seventh problem in the list was titled, “irrationality and transcendence of certain numbers”. In this, he raised the question whether an irrational logarithm of an algebraic number to an algebraic base is transcendental. This can be asked differently as whether an irrational quotient of natural logarithms of algebraic numbers is transcendental or whether $\alpha ^\beta$ is transcendental for any algebraic number $\ne 0,1$ and any algebraic irrational $\beta .$ The question about properties of $\log _\alpha \beta$ with $\alpha ,\beta$ rational was stated by Euler. He supposed that they are transcendental with exceptions like $\log _2 4$ but no proofs were given. Although this problem has similarity with the Hermite–Lindemann–Weierstrass theorem, Hilbert felt that it is extraordinarily difficult. He was of the view that proof of the irrationality of $2^{\sqrt{2}}$ belongs to the more distant future than the proof of Riemann Hypothesis or Fermat’s Last Theorem. He was mistaken.

The powers of the mind are like the rays of the sun when they are concentrated they illumine

—Vivekananda