Hilbert's seventh problem concerns the irrationality and transcendence of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen). In its geometric formulation, it asks whether the following statement is provably true:

In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, then the ratio between base and side is always transcendental.

A special case of this problem asks:

Is a^b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b?

When b is rational, a^b will be algebraic.

The special problem was solved by Aleksandr Gelfond in 1934, and refined by Theodor Schneider (1911 - ) in 1935. They proved that a^b is transcendental when b is both algebraic and irrational. This result is known as Gelfond's theorem or the Gelfond-Schneider theorem.

From the point of view of generalisations, this is the case

blog (α) + log(β) = 0

of the general linear form in logarithms .

See also:

• Alan Baker
• Gelfond's conjecture
• Hilbert's problems