Hilbert's seventh problem. In 1900 David Hilbert announced a list of twenty-three outstanding unsolved problems. The seventh problem was settled by the publication of the following result in 1934 by A. O. Gelfond, which was followed by an independent proof by Th. Schneider in 1935.

Theorem 10.1. If α and β are algebraic numbers with α ≠ 0, α ≠ 1, and if β is not a real rational number, then any value of αβ is transcendental.

Remarks. The hypothesis that “β is not a real rational number” is usually stated in the form “β is irrational.” Our wording is an attempt to avoid the suggestion that β must be a real number. Such a number as β = 2 + 3i, sometimes called a “complex rational number,” satisfies the hypotheses of the theorem. Thus the theorem establishes the transcendence of such numbers as 2i. In general, αβ = exp {β log α) is multiplevalued, and this is the reason for the phrase “any value of” in the statement of Theorem 10.1. One value of i−2i = exp {−2i log i} is eπ, and so this is transcendental according to the theorem.

Before proceeding to the proof of Theorem 10.1, we state an alternative form of the result.