Hilbert's seventh problem


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.

Background and the statement

In 1919, Hilbert gave a lecture on number theory and spoke of three conjectures: the Riemann hypothesis, Fermat's Last Theorem, and the transcendence of. He mentioned to the audience that he did not expect anyone in the hall to live long enough to see proof of this result. In his seventh problem, under the title "Irrationality and Transcendence of Certain Numbers", two specific equivalent questions were asked:
  1. In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, is the ratio between base and side always transcendental?
  2. Is always transcendental, for algebraic and irrational algebraic ?
For preliminary, an algebraic number is the root of a non-zero polynomial in one variable with rational coefficients, and a transcendental number is the opposite of an algebraic number. An irrational number is a number that cannot be expressed by the fraction of two integers; the denominator cannot be zero.
In addressing the second question more specifically, Hilbert asked about the transcendence and irrationality of the numbers and. The first number is known as the Gelfond–Schneider constant or the Hilbert number.

Solution

The proof regarding the transcendence of was published by Kuzmin in 1930, well within Hilbert's own lifetime. Namely, Kuzmin proved the case where the exponent is a real quadratic irrational.
In 1934, Aleksandr Gelfond and Theodor Schneider proved more generally, by extending the number to an arbitrary algebraic irrational. Respectively, they independently answered the second problem in the affirmative and refined it. This result is known as Gelfond's theorem or the Gelfond–Schneider theorem. From the generalization's point of view, this is the case
of the general linear form in logarithms, which was studied by Gelfond and then solved by Alan Baker, who was awarded a Fields Medal in 1970 for this achievement. The result is called the Gelfond conjecture or Baker's theorem.