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 (Irrationalität und Transzendenz bestimmter Zahlen).

Statement of the problem

Two specific equivalent[1] 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 $a^{b}$ always transcendental, for algebraic $a\not \in \{0,1\}$ and irrational algebraic $b$ ?

Solution

The question (in the second form) 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 $a^{b}$ is algebraic for algebraic a and rational b.)

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

b\ln {\alpha }+\ln {\beta }=0

of the general linear form in logarithms which was studied 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.

References

Feldman, N. I.; Nesterenko, Yu. V. (1998). Parshin, A. N.; Shafarevich, I. R. (eds.). Transcendental Numbers. Number Theory IV. Springer-Verlag Berlin Heidelberg. pp. 146–147. ISBN 978-3-540-61467-8.

Bibliography

Tijdeman, Robert (1976). "On the Gel'fond–Baker method and its applications". In Felix E. Browder (ed.). Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. XXVIII.1. American Mathematical Society. pp. 241–268. ISBN 978-0-8218-1428-4. Zbl 0341.10026.
Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. 49 (Second ed.). p. 61. ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002.

External links

English translation of Hilbert's original address

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[1] Feldman, N. I.; Nesterenko, Yu. V. (1998). Parshin, A. N.; Shafarevich, I. R. (eds.). Transcendental Numbers. Number Theory IV. Springer-Verlag Berlin Heidelberg. pp. 146–147. ISBN 978-3-540-61467-8.