Alexander Gelfond

Alexander Osipovich Gelfond (Russian: Александр Осипович Гелфонд; October 24 1906, St Petersburg - November 7 1968, Moscow) was a Russian mathematician, author of the Gelfond's theorem.

Biography

Alexander Gelfond was born in Petrograd (currently Saint-Petersbug) in a family of a professional physician and an amateur philosopher Osip Isaakovich Gelfond. He entered the Moscow State University in 1924, started his postgraduate studies there in 1927 and obtained his PhD in 1930. His advisers were Alexander Khinchin and Vyacheslav Stepanov.

In 1930 he worked for five months in Germany (in Berlin and Göttingen) were he worked with Edmund Landau, Carl Ludwig Siegel and David Hilbert. In 1931 he started teaching as a Professor at the Moscow State University and worked there until the last day of his life. Since 1933 he also worked at the Steklov Institute of Mathematics. In 1939 he was elected a Corresponding member of the Academy of Sciences of the USSR.

Results

Alexander Gelfond obtained important results in several mathematical domains including number theory, analytic functions, integral equations and the history of mathematics, but his most famous result is his eponymous theorem:

If

\alpha

and

\beta

are algebraic numbers (with

\alpha \neq 0

and

\alpha \neq 1

), and if

\beta

is not a real rational number, then any value of

\alpha ^{\beta }

is a transcendental number.

This is the famous 7th Hilbert's problem. Gelfond proved a special case of the theorem in 1929, when he was a postgraduate student and fully proved it in 1934. In 1935 the same theorem was independently proved by Theodor Schneider and so the theorem is often known as the Gelfond–Schneider theorem. In 1929 Gelfond proposed an extension of the theorem known as the Gelfond's conjecture that was proved by Alan Baker in 1966.

Before Gelfond's works only a few numbers such as e and π were known to be transcendental. After his works an infinite number of transcendentals could be easily obtained. Some of them a named in Gelfond's honor:

$2^{\sqrt {2}}$ is known as the Gelfond–Schneider constant
$e^{\pi }\,$ is known as Gelfond's constant

References

Alexander Gelfond at the Mathematics Genealogy Project
O'Connor, John J.; Robertson, Edmund F., "Alexander Gelfond", MacTutor History of Mathematics Archive, University of St Andrews
B.V. Levin, N.I. Feldman, A.B. Sidlovsky (1971). "Alexander O. Gelfond". Acta Arithmetica. 17: 315–336.{{cite journal}}: CS1 maint: multiple names: authors list (link)