Gelfond's Theorem

The famous mathematician David Hilbert proposed a list of twenty-three questions at the Second International Congress on Mathematics in 1900. The seventh of these was:

If a is algebraic and not equal to 1 or 0, and b is irrational, what is known about the transcendentality of a^b?

In 1934 Aleksander Gelfond partially solved this problem for the case that b is algebraic. Formally stated, Gelfond's Theorem reads:

For all a ∈ A, a ≠ 1, 0 and b ∈ A and b ∉ Q, a^b ∉ A.

Here A is the set of all algebraic numbers (those which are the solutions of finite polynomials with integer coefficients) and Q is the set of all rational numbers (those which are the ratios of two integers.) One famous use of this theorem is to show that e^π is transcendental, as follows.

-1 ∈ A, ≠ 1, 0
-i ∈ A, ∉ Q
∴ (-1)^(-i) ∉ A

e^(iπ) = -1
(-1)^(-i) = (e^(iπ))^(-i) = e^(i*-i*π) = e^π
∴ e^π ∉ A

e^π is transcendental.

The general case posed in Hilbert's seventh problem remains unsolved.

Sources

Eric W. Weisstein. Eric Weisstein's World of Mathematics. mathworld.wolfram.com. Wolfram Research, Inc. Accessed February 4th, 2003.