# Aleksandr Osipovich Gelfond

# Aleksandr Osipovich Gelfond

**1906-1968**

**Russian Mathematician**

In 1929 and 1934 Aleksandr Gelfond published papers on his inquiries into transcendental numbers, or numbers that are not the solution to an algebraic equation with rational coefficients. He was specifically concerned with Hilbert's seventh problem, which revolved around the assumption that *a*^{b} is transcendental if *a* is any algebraic number other than 0 or 1 and *b* is any irrational algebraic number. Using linear forms of exponential functions, he solved the problem and established what became known as Gelfond's theorem.

Gelfond was born in 1906 in St. Petersburg, Russia, which would later become Leningrad, U.S.S.R. His father, a physician with an interest in philosophy, was an acquaintance of Lenin, who in 1917 became leader of the new Communist state. In 1924, the year Lenin died, Gelfond entered Moscow University, where he completed his undergraduate studies in mathematics three years later. He performed his postgraduate studies between 1927 and 1930, at which point he took a position at Moscow Technological College. A year later, he became a professor of mathematics at Moscow University, where he remained until his death.

An ardent student of mathematical history, Gelfond was intrigued by a proposition made by Leonhard Euler (1707-1783) in 1748 stating
that logarithms of rational numbers with rational bases are either rational or transcendental. David Hilbert (1862-1943), who in 1900 presented the mathematical community with 23 problems that would provoke debate for decades to come, had built on Euler's conjecture by proposing, in his seventh problem, that *a*^{b} is transcendental if *a* is any algebraic number other than 0 or 1 and *b* is any irrational algebraic number. In his 1929 paper, Gelfond drew connections between the arithmetic nature of a number's values and the properties of an analytic function. He followed this paper with a 1934 work on Hilbert's seventh problem, in which he showed that *a*^{b} is indeed a transcendental number—a statement that came to be known as Gelfond's theorem.

Concurrent with his teaching at Moscow University, Gelfond received a post at the Soviet Academy of Sciences Mathematical Institute in 1933, and in 1935 he earned his doctorate in mathematics and physics. Those times were frightening ones in the Soviet Union, with the atmosphere of ever-present terror created by Stalin's dictatorship, but Gelfond—a Communist Party member—kept a low profile.

During the 1930s, Gelfond was peripherally associated with the Luzitania, a coterie of students and admirers who gathered around the mathematician Nikolai Luzin (1883-1950). Though Luzin was later charged with ideological sabotage and nearly subjected to a trial before receiving a pardon, Gelfond himself escaped censure. He risked endangering himself, however, for Ilya Piatetski-Shapiro, a Jewish student denied admission to the Moscow University graduate school in the wave of Stalin-inspired anti-Semitic hysteria that followed World War II. Piatetski-Shapiro remained a devoted admirer, and was with Gelfond when he died on November 7, 1968, in a Moscow hospital.

**JUDSON KNIGHT**

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**Aleksandr Osipovich Gelfond**