The Hilbert's seventh problem reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Hilbert's seventh problem

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

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