Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen).
Contents
Statement of the problem
Two specific questions are asked:
- In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, is then the ratio between base and side always transcendental?
- Is
a b a ∉ { 0 , 1 } and irrational algebraicb ?
Solution
The second question was answered in the affirmative by Aleksandr Gelfond in 1934, and refined by Theodor Schneider in 1935. This result is known as Gelfond's theorem or the Gelfond–Schneider theorem. (The restriction to irrational b is important, since it is easy to see that
From the point of view of generalisations, this is the case
of the general linear form in logarithms which was attacked by Gelfond and then solved by Alan Baker. It is called the Gelfond conjecture or Baker's theorem. Baker was awarded a Fields Medal in 1970 for this achievement.
The first question is a consequence of the second question.