The List of Hilbert’s Twenty-Three Problems: Hilbert’s Seventh Problem

Irrationality and Transcendence of Certain Numbers: Is $a^b$ transcendental, for algebraic $a ≠ 0,1$ and irrational algebraic $b$

A number is called algebraic if it can be the zero of a polynomial with rational coefficients. 

For example, $2$ is a zero of the polynomial $x − 2$, and $√2$ is a zero of the polynomial $x^2 − 2$. Algebraic numbers can be either rational or irrational; transcendental numbers like $π$ are irrational numbers that are not algebraic. Hilbert’s seventh problem concerns powers of algebraic numbers. 

Consider the expression $a^b$, where $a$ is an algebraic number other than $0$ or $1$ and $b$ is an irrational algebraic number. Must $a^b$ be transcendental? 

In $1934$, Aleksandr Gelfond showed that the answer is yes.

Πηγή: abakcus