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Hans Julius Zassenhaus (28 May 1912 – 21 November 1991) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, known for work in many parts of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, and as a pioneer of
computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expression ...
.


Biography

He was born in
Koblenz Koblenz (; Moselle Franconian: ''Kowelenz''), spelled Coblenz before 1926, is a German city on the banks of the Rhine and the Moselle, a multi-nation tributary. Koblenz was established as a Roman military post by Drusus around 8 B.C. Its na ...
in 1912. His father was a
historian A historian is a person who studies and writes about the past and is regarded as an authority on it. Historians are concerned with the continuous, methodical narrative and research of past events as relating to the human race; as well as the st ...
and advocate for Reverence for Life as expressed by
Albert Schweitzer Ludwig Philipp Albert Schweitzer (; 14 January 1875 – 4 September 1965) was an Alsatian-German/French polymath. He was a theologian, organist, musicologist, writer, humanitarian, philosopher, and physician. A Lutheran minister, Schwei ...
. Hans had two brothers, Guenther and Wilfred, and sister Hiltgunt, who wrote an autobiography in 1974. According to her, their father lost his position as school principal due to his philosophy. She wrote:
Hiltgunt Zassenhaus Hiltgunt Margret Zassenhaus (10 July 1916 – 20 November 2004) was a German philologist who worked as an interpreter in Hamburg, Germany during World War II, and later as a physician in the United States. She was honoured for her efforts to a ...
(1974) ''Walls: Resisting the Third Reich'',
Beacon Press Beacon Press is an American left-wing non-profit book publisher. Founded in 1854 by the American Unitarian Association, it is currently a department of the Unitarian Universalist Association. It is known for publishing authors such as James ...
:Hans, my eldest brother, studied mathematics. My brothers Guenther and Wilfred were in medical school. ... only students who participated in Nazi activities would get scholarships. That left us out. Together we made an all-out effort. ... soon our house became a beehive. Day in and day out for the next four years a small army of children of all ages would arrive to be tutored. At the
University of Hamburg The University of Hamburg (german: link=no, Universität Hamburg, also referred to as UHH) is a public research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ('' Allgemeines Vor ...
Zassenhaus came under the influence of
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing ...
. As he wrote later: :His introductory course in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
that I attended at the age of 17 converted me from a
theoretical physicist Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
to a mathematician. When just 21, Zassenhaus was studying
composition series In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natu ...
in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. He proved his butterfly lemma that provides a refinement of two normal chains to isomorphic central chains. Inspired by Artin, Zassenhaus wrote a
textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textboo ...
''Lehrbuch der Gruppentheorie'' that was later translated as ''Theory of Groups''. His thesis was on doubly transitive permutation groups with Frobenius groups as stabilizers. These groups are now called Zassenhaus groups. They have had a deep impact on the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or els ...
. He obtained his doctorate in June 1934, and took the teachers’ exam the next May. He became a scientific assistant at
University of Rostock The University of Rostock (german: link=no, Universität Rostock) is a public university located in Rostock, Mecklenburg-Vorpommern, Germany. Founded in 1419, it is the third-oldest university in Germany. It is the oldest university in contine ...
. In 1936 he became assistant to Artin back in Hamburg, but Artin departed for the USA the following year. Zassenhaus gave his
Habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including ...
in 1938. According to his sister Hiltgunt, Hans was "called up as a research scientist at a weather station" for his part in the German war effort. Zassenhaus married Lieselotte Lohmann in 1942. The couple raised three children: Michael (born 1943), Angela (born 1947), and Peter (born 1949). In 1943 Zassenhaus became extraordinary
professor Professor (commonly abbreviated as Prof.) is an academic rank at universities and other post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who professes". Professor ...
. He became Managing Director of the Hamburg Mathematical Seminar. After the war, and as a fellow of the
British Council The British Council is a British organisation specialising in international cultural and educational opportunities. It works in over 100 countries: promoting a wider knowledge of the United Kingdom and the English language (and the Welsh la ...
, Zassenhaus visited the
University of Glasgow , image = UofG Coat of Arms.png , image_size = 150px , caption = Coat of arms Flag , latin_name = Universitas Glasguensis , motto = la, Via, Veritas, Vita , ...
in 1948. There he was given an honorary Master of Arts degree. The following year he joined the faculty of
McGill University McGill University (french: link=no, Université McGill) is an English-language public research university located in Montreal, Quebec, Canada. Founded in 1821 by royal charter granted by King George IV,Frost, Stanley Brice. ''McGill Universit ...
where the endowments of Peter Redpath financed a professorship. He was at McGill for a decade with
leaves of absence The labour law concept of leave, specifically paid leave or, in some countries' long-form, a leave of absence, is an authorised prolonged absence from work, for any reason authorised by the workplace. When people "take leave" in this way, they are ...
to the
Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
(1955/6) and
California Institute of Technology The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...
(1958/9). There he was using computers to advance
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
. In 1959 Zassenhaus began teaching at
University of Notre Dame The University of Notre Dame du Lac, known simply as Notre Dame ( ) or ND, is a private Catholic university, Catholic research university in Notre Dame, Indiana, outside the city of South Bend, Indiana, South Bend. French priest Edward Sorin fo ...
and became director of its computing center in 1964. Zassenhaus was a Mershon visiting professor at
Ohio State University The Ohio State University, commonly called Ohio State or OSU, is a public land-grant research university in Columbus, Ohio. A member of the University System of Ohio, it has been ranked by major institutional rankings among the best pu ...
in the fall of 1963. In 1965 he came to Ohio State permanently. The mathematics department was led by Arnold Ross; Zassenhaus found a home there until his retirement in 1982. Nonetheless, he continued to take leaves of absence for visits to
Göttingen Göttingen (, , ; nds, Chöttingen) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. At the end of 2019, t ...
(summer 1967),
Heidelberg Heidelberg (; Palatine German language, Palatine German: ''Heidlberg'') is a city in the States of Germany, German state of Baden-Württemberg, situated on the river Neckar in south-west Germany. As of the 2016 census, its population was 159,914 ...
(summer 1969),
UCLA The University of California, Los Angeles (UCLA) is a public university, public Land-grant university, land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a Normal school, teachers colle ...
(fall 1970),
Warwick Warwick ( ) is a market town, civil parish and the county town of Warwickshire in the Warwick District in England, adjacent to the River Avon. It is south of Coventry, and south-east of Birmingham. It is adjoined with Leamington Spa and W ...
(fall 1972), CIT (1974/75), U Montreal (1977/78),
Saarbrücken Saarbrücken (; french: link=no, Sarrebruck ; Rhine Franconian: ''Saarbrigge'' ; lb, Saarbrécken ; lat, Saravipons, lit=The Bridge(s) across the Saar river) is the capital and largest city of the state of Saarland, Germany. Saarbrücken is ...
(1979/80). He served as editor in chief of the Journal of Number Theory from its first issue in 1967. He won a Lester R. Ford Award in 1968. Hans Zassenhaus died in
Columbus, Ohio Columbus () is the state capital and the most populous city in the U.S. state of Ohio. With a 2020 census population of 905,748, it is the 14th-most populous city in the U.S., the second-most populous city in the Midwest, after Chicago, an ...
on November 21, 1991. His doctoral students include Joachim Lambek.


Important publications

* Hans Julius Zassenhaus (1937), ''Lehrbuch der Gruppentheorie'' ("Textbook of group theory"), 2nd edition (1960),''The theory of groups''. A famous
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
book based on a course by
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing ...
given at the
University of Hamburg The University of Hamburg (german: link=no, Universität Hamburg, also referred to as UHH) is a public research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ('' Allgemeines Vor ...
during winter semester 1933 and summer semester 1934. * Zassenhaus showed that there are just seven
near-field Near field may refer to: * Near-field (mathematics), an algebraic structure * Near-field region, part of an electromagnetic field * Near field (electromagnetism) ** Magnetoquasistatic field, the magnetic component of the electromagnetic near f ...
s that are not division rings or Dickson near-fields in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 11, pp 187–220. * In 1977
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes refere ...
published ''Number Theory and Algebra'', a collection of papers dedicated to Henry B. Mann, Arnold E. Ross, and Olga Taussky-Todd, edited by Zassenhaus (). It included "A Theorem on Cyclic Algebras" by Zassenhaus. *
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...
published ''Algorithmic Algebraic Number Theory'' written by Zassenhaus and M. Pohst in 1989 (). A second edition appeared in 1993. * . The paper that introduced the
Cantor–Zassenhaus algorithm In computational algebra, the Cantor–Zassenhaus algorithm is a method for factoring polynomials over finite fields (also called Galois fields). The algorithm consists mainly of exponentiation and polynomial GCD computations. It was invented by ...
for factoring polynomials.


See also

*
Pfister's sixteen-square identity In algebra, Pfister's sixteen-square identity is a non- bilinear identity of form \left(x_1^2+x_2^2+x_3^2+\cdots+x_^2\right)\left(y_1^2+y_2^2+y_3^2+\cdots+y_^2\right) = z_1^2+z_2^2+z_3^2+\cdots+z_^2 It was first proven to exist by H. Zassenhaus a ...
* Z-group * Zassenhaus dual expansion * Zassenhaus algorithm * Zassenhaus formula * Zassenhaus group * Zassenhaus lemma * Zassenhaus neghbourhood *
Berlekamp–Zassenhaus algorithm In mathematics, in particular in computer algebra, computational algebra, the Berlekamp–Zassenhaus algorithm is an algorithm for factoring polynomials over the integers, named after Elwyn Berlekamp and Hans Zassenhaus. As a consequence of Gau ...
*
Cantor–Zassenhaus algorithm In computational algebra, the Cantor–Zassenhaus algorithm is a method for factoring polynomials over finite fields (also called Galois fields). The algorithm consists mainly of exponentiation and polynomial GCD computations. It was invented by ...
* Schur–Zassenhaus theorem


References

* M. Pohst (1994
"Hans Zassenhaus"
Journal of Number Theory 47:1–19.


External links

*
Biography from the Ohio State University
{{DEFAULTSORT:Zassenhaus, Hans Julius 1912 births 1991 deaths 20th-century German mathematicians Group theorists University of Notre Dame faculty Ohio State University faculty