**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*, where

^{b}*a*is an algebraic number other than 0 or 1 and

*b*is an irrational algebraic number. Must

*a*be transcendental? In 1934, Aleksandr Gelfond showed that the answer is yes.

^{b}