Alexander Gelfond
Encyclopedia
Alexander Osipovich Gelfond ' onMouseout='HidePop("26106")' href="/topics/Moscow">Moscow
) was a Soviet mathematician, author of Gelfond's theorem.
in the family of a professional physician
and 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 advisors were Alexander Khinchin
and Vyacheslav Stepanov.
In 1930 he stayed for five months in Germany
(in Berlin
and Göttingen
) where 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 for his works in the field of Cryptography
. According to Vladimir Arnold
, during World War II
Gelfond was the Chief Cryptographer of the Soviet Navy
, analytic functions, integral equation
s and the history of mathematics
, but his most famous result is his eponymous theorem:
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 are named in Gelfond's honor:
Moscow
Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...
) was a Soviet mathematician, author of Gelfond's theorem.
Biography
Alexander Gelfond was born in St Petersburg, Russian EmpireRussian Empire
The Russian Empire was a state that existed from 1721 until the Russian Revolution of 1917. It was the successor to the Tsardom of Russia and the predecessor of the Soviet Union...
in the family of a professional physician
Physician
A physician is a health care provider who practices the profession of medicine, which is concerned with promoting, maintaining or restoring human health through the study, diagnosis, and treatment of disease, injury and other physical and mental impairments...
and amateur philosopher Osip Isaakovich Gelfond. He entered the Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
in 1924, started his postgraduate studies there in 1927 and obtained his PhD
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
in 1930. His advisors were Alexander Khinchin
Aleksandr Yakovlevich Khinchin
Aleksandr Yakovlevich Khinchin was a Soviet mathematician and one of the most significant people in the Soviet school of probability theory. He was born in the village of Kondrovo, Kaluga Governorate, Russian Empire. While studying at Moscow State University, he became one of the first followers...
and Vyacheslav Stepanov.
In 1930 he stayed for five months in Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
(in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
and Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
) where he worked with Edmund Landau
Edmund Landau
Edmund Georg Hermann Landau was a German Jewish mathematician who worked in the fields of number theory and complex analysis.-Biography:...
, Carl Ludwig Siegel
Carl Ludwig Siegel
Carl Ludwig Siegel was a mathematician specialising in number theory and celestial mechanics. He was one of the most important mathematicians of the 20th century.-Biography:...
and David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
. 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
Steklov Institute of Mathematics
Steklov Institute of Mathematics or Steklov Mathematical Institute is a research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. It was established April 24, 1934 by the decision of the General Assembly of the Academy of Sciences of the USSR in...
.
In 1939 he was elected a Corresponding member of the Academy of Sciences of the USSR for his works in the field of Cryptography
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
. According to Vladimir Arnold
Vladimir Arnold
Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...
, during World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
Gelfond was the Chief Cryptographer of the Soviet Navy
Soviet Navy
The Soviet Navy was the naval arm of the Soviet Armed Forces. Often referred to as the Red Fleet, the Soviet Navy would have played an instrumental role in a Warsaw Pact war with NATO, where it would have attempted to prevent naval convoys from bringing reinforcements across the Atlantic Ocean...
Results
Gelfond obtained important results in several mathematical domains including number theoryNumber theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, analytic functions, integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
s and the history of mathematics
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
, but his most famous result is his eponymous theorem:
- If α and β are algebraic numberAlgebraic numberIn mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
s (with α≠0 and α≠1), and if β is not a realReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
rational numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
, then any value of αβ is a transcendental numberTranscendental numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
.
This is the famous 7th Hilbert's problem
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...
. 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
Theodor Schneider
Theodor Schneider was a German mathematician, best known for providing proof of what is now known as the Gelfond–Schneider theorem in 1935....
and so the theorem is often known as the Gelfond–Schneider theorem
Gelfond–Schneider theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider...
. 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
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
and π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
were known to be transcendental. After his works an infinite number of transcendentals could be easily obtained. Some of them are named in Gelfond's honor:
-
is known as the Gelfond–Schneider constant -
is known as Gelfond's constantGelfond's constantIn mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e to the power of π. Like both e and π, this constant is a transcendental number. This can be proven by the Gelfond–Schneider theorem and noting the fact that...
.