Robert Remak (mathematician)
Encyclopedia
Robert Erich Remak was a German mathematician. He is chiefly remembered for his work in group theory
(Remak decomposition). His other interests included algebraic number theory
, mathematical economics
and geometry of numbers
. Robert Remak was the son of the neurologist Ernst Julius Remak.
under Ferdinand Georg Frobenius
and received his doctorate in 1911. His dissertation, Über die Zerlegung der endlichen Gruppen in indirekte unzerlegbare Faktoren ("On the decomposition of a finite group into indirect indecomposable factors") established that any two decompositions of a finite group
into a direct product
are related by a central automorphism. A weaker form of this statement, uniqueness, was first proved by Joseph Wedderburn
in 1909. Later the theorem was generalized by Wolfgang Krull
and Otto Schmidt
to some classes of infinite groups and became known as the Krull–Schmidt theorem
or the Krull–Remak–Schmidt theorem.
Although the dissertation was first first submitted in 1911, it was rejected several times and Remak did not obtain his Habilitation
until 1929. In the meantime, he wrote several papers on the geometry of numbers
. Between 1929 and 1933 Remak lectured as a Privatdozent
at Humboldt University. In the 1929 essay Kann die Volkwirtschaftslehre eine exakte Wissenschaft werden? ("Can economics become an exact science?"), Remak analyzed price formation in socialist and capitalist economies. He also anticipated the role played by digital computer
s in numerical solution of systems of linear equations. Remak's analysis may have influenced John von Neumann
, who was a fellow lecturer in Berlin, but most of it has not been translated into English and it remains little known and appreciated in the English-speaking world. In 1932 Remak published a paper giving a lower bound for the regulator of an algebraic number field
in terms of the numbers r1 and r2 of real embeddings and pairs of complex embeddings. He went on to investigate relations between the regulator and the discriminant of an algebraic number field
, isolating an important class of CM-field
s ("fields with unit defect"). His last two papers on the subject appeared in Compositio Mathematica in 1952 and 1954, more than ten years after his death.
After the Nazis seized power
in 1933 and the Civil Service Law
was passed a few months later, Remak, who was of Jewish ancestry, lost his right to teach in September 1933. He was arrested on Kristallnacht
, November 9, 1938 and was interned at Sachsenhausen concentration camp
for several weeks. After an unsuccessful campaign by his wife to secure a permission for him to emigrate to the United States, he was released and permitted to leave for Amsterdam
. In 1942, he was arrested by the German occupational authorities in the Netherlands and deported to Auschwitz, where he perished.
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
(Remak decomposition). His other interests included algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
, mathematical economics
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...
and geometry of numbers
Geometry of numbers
In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
. Robert Remak was the son of the neurologist Ernst Julius Remak.
Biography
Robert Remak studied at Humboldt University of BerlinHumboldt University of Berlin
The Humboldt University of Berlin is Berlin's oldest university, founded in 1810 as the University of Berlin by the liberal Prussian educational reformer and linguist Wilhelm von Humboldt, whose university model has strongly influenced other European and Western universities...
under Ferdinand Georg Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
and received his doctorate in 1911. His dissertation, Über die Zerlegung der endlichen Gruppen in indirekte unzerlegbare Faktoren ("On the decomposition of a finite group into indirect indecomposable factors") established that any two decompositions of a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
into a direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
are related by a central automorphism. A weaker form of this statement, uniqueness, was first proved by Joseph Wedderburn
Joseph Wedderburn
Joseph Henry Maclagan Wedderburn was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras...
in 1909. Later the theorem was generalized by Wolfgang Krull
Wolfgang Krull
Wolfgang Krull was a German mathematician working in the field of commutative algebra.He was born in Baden-Baden, Imperial Germany and died in Bonn, West Germany.- See also :* Krull dimension* Krull topology...
and Otto Schmidt
Otto Schmidt
Otto Yulyevich Schmidt was a Soviet scientist, mathematician, astronomer, geophysicist, statesman, academician, Hero of the USSR , and member of the Communist Party.-Biography:He was born in Mogilev, Russian Empire...
to some classes of infinite groups and became known as the Krull–Schmidt theorem
Krull–Schmidt theorem
In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.-Definitions:...
or the Krull–Remak–Schmidt theorem.
Although the dissertation was first first submitted in 1911, it was rejected several times and Remak did not obtain his Habilitation
Habilitation
Habilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...
until 1929. In the meantime, he wrote several papers on the geometry of numbers
Geometry of numbers
In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
. Between 1929 and 1933 Remak lectured as a Privatdozent
Privatdozent
Privatdozent or Private lecturer is a title conferred in some European university systems, especially in German-speaking countries, for someone who pursues an academic career and holds all formal qualifications to become a tenured university professor...
at Humboldt University. In the 1929 essay Kann die Volkwirtschaftslehre eine exakte Wissenschaft werden? ("Can economics become an exact science?"), Remak analyzed price formation in socialist and capitalist economies. He also anticipated the role played by digital computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
s in numerical solution of systems of linear equations. Remak's analysis may have influenced John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
, who was a fellow lecturer in Berlin, but most of it has not been translated into English and it remains little known and appreciated in the English-speaking world. In 1932 Remak published a paper giving a lower bound for the regulator of an algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
in terms of the numbers r1 and r2 of real embeddings and pairs of complex embeddings. He went on to investigate relations between the regulator and the discriminant of an algebraic number field
Discriminant of an algebraic number field
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the algebraic number field...
, isolating an important class of CM-field
CM-field
In mathematics, a CM-field is a particular type of number field K, so named for a close connection to the theory of complex multiplication. Another name used is J-field. Specifically, K is a quadratic extension of a totally real field which is totally imaginary, i.e...
s ("fields with unit defect"). His last two papers on the subject appeared in Compositio Mathematica in 1952 and 1954, more than ten years after his death.
After the Nazis seized power
Machtergreifung
Machtergreifung is a German word meaning "seizure of power". It is normally used specifically to refer to the Nazi takeover of power in the democratic Weimar Republic on 30 January 1933, the day Hitler was sworn in as Chancellor of Germany, turning it into the Nazi German dictatorship.-Term:The...
in 1933 and the Civil Service Law
Law for the Restoration of the Professional Civil Service
The Law for the Restoration of the Professional Civil Service , also known as Civil Service Law, Civil Service Restoration Act, and Law to Re-establish the Civil Service, was a law passed by the National Socialist regime on April 7, 1933, two months after Adolf...
was passed a few months later, Remak, who was of Jewish ancestry, lost his right to teach in September 1933. He was arrested on Kristallnacht
Kristallnacht
Kristallnacht, also referred to as the Night of Broken Glass, and also Reichskristallnacht, Pogromnacht, and Novemberpogrome, was a pogrom or series of attacks against Jews throughout Nazi Germany and parts of Austria on 9–10 November 1938.Jewish homes were ransacked, as were shops, towns and...
, November 9, 1938 and was interned at Sachsenhausen concentration camp
Sachsenhausen concentration camp
Sachsenhausen or Sachsenhausen-Oranienburg was a Nazi concentration camp in Oranienburg, Germany, used primarily for political prisoners from 1936 to the end of the Third Reich in May, 1945. After World War II, when Oranienburg was in the Soviet Occupation Zone, the structure was used as an NKVD...
for several weeks. After an unsuccessful campaign by his wife to secure a permission for him to emigrate to the United States, he was released and permitted to leave for Amsterdam
Amsterdam
Amsterdam is the largest city and the capital of the Netherlands. The current position of Amsterdam as capital city of the Kingdom of the Netherlands is governed by the constitution of August 24, 1815 and its successors. Amsterdam has a population of 783,364 within city limits, an urban population...
. In 1942, he was arrested by the German occupational authorities in the Netherlands and deported to Auschwitz, where he perished.
External links
- Willy Tiabou, Christoph Bichlmeier: Verfolgte Mathematiker