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. ^[1]

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 ${\displaystyle 2^{\sqrt {2}}}$. He mentioned to the audience that he did not expect anyone in the hall to live long enough to see proof of this result.^{[ citation needed ]} In his seventh problem, under the title "Irrationality and Transcendence of Certain Numbers" (Irrationalität und Transzendenz bestimmter Zahlen in German), ^[1] two specific equivalent questions were asked: ^[2]

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 ${\displaystyle a^{b}}$ always transcendental, for algebraic ${\displaystyle a\not \in \{0,1\}}$ and irrational algebraic ${\displaystyle b}$?

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 (i.e., not the root of a polynomial). 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 ${\displaystyle 2^{\sqrt {2}}}$ and ${\displaystyle e^{\pi }=i^{-2i}}$. The first number is known as the Gelfond–Schneider constant or the Hilbert number. ^[1]

Solution

Main article: Gelfond–Schneider theorem

The proof regarding the transcendence of ${\displaystyle 2^{\sqrt {2}}}$ was published by Kuzmin in 1930, well within Hilbert's own lifetime. Namely, Kuzmin proved the case where the exponent ${\displaystyle b}$ is a real quadratic irrational. ^[3]

In 1934, Aleksandr Gelfond and Theodor Schneider proved more generally, by extending the number ${\displaystyle b}$ 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. ^{[4] [5]} From the generalization's point of view, this is the case $${\displaystyle b\ln {\alpha }+\ln {\beta }=0}$$ of the general linear form in logarithms, which was studied by Gelfond^{[ citation needed ]} and then solved by Alan Baker, ^[6] who was awarded a Fields Medal in 1970 for this achievement. The result is called the Gelfond conjecture or Baker's theorem.

References

1. Hilbert, David (1902). "Mathematical Problems". Bulletin of the American Mathematical Society. 8 (10): 437–479. doi: 10.1090/S0002-9904-1902-00923-3 . MR   1557926. Earlier publications (in the original German) appeared in Göttinger Nachrichten, 1900, pp. 253–297, and Archiv der Mathematik und Physik, 3rd series, vol. 1 (1901), pp. 44-63, 213–237.
2. 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.
3. Kuzmin, R. O. (1930). "On a new class of transcendental numbers". Izvestiya Akademii Nauk, Seriya Matematicheskaya. 7: 585–597.
4. • Gelfond, Aleksandr (1934). "Sur le septième problème de D. Hilbert". Comptes Rendus de l'Académie des Sciences de l'URSS (in Russian and French). 2 (2): 1–6. JFM   60.0163.04.
   • Gelfond, Aleksandr (1934). "Sur le septième Problème de Hilbert". Bulletin de l'Académie des Sciences de l'URSS, Classe des Sciences Mathématiques et Naturelles. Série VII (4): 623–634. JFM   60.0164.01.
   • Gelfond, Aleksandr (1934). "On the seventh problem of D. Hilbert" [Sur le septième problème de D. Hilbert.]. Comptes Rendus de l'Académie des Sciences de l'URSS (in Russian and French). 2: 1–3, 4–6. Zbl   0009.05302.
5. Schneider, Theodor (1934). "Transzendenzuntersuchungen periodischer Funktionen. I. Transzendenz von Potenzen". Journal für die Reine und Angewandte Mathematik. 172: 65–69. Zbl   0010.10501.
6. • Baker, Alan (1966). "Linear forms in the logarithms of algebraic numbers. I". Mathematika . 13 (2): 204–216. doi:10.1112/S0025579300003971. ISSN   0025-5793. MR   0220680.
   • Baker, Alan (1967a). "Linear forms in the logarithms of algebraic numbers. II". Mathematika . 14: 102–107. doi:10.1112/S0025579300008068. ISSN   0025-5793. MR   0220680.
   • Baker, Alan (1967b). "Linear forms in the logarithms of algebraic numbers. III". Mathematika . 14 (2): 220–228. doi:10.1112/S0025579300003843. ISSN   0025-5793. MR   0220680.

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. Vol. 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. Vol. 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