Moderne Algebra
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''Moderne Algebra'' is a two-volume German textbook on graduate
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 term ''a ...
by , originally based on lectures given 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 lar ...
in 1926 and by from 1924 to 1928. The English translation of 1949–1950 had the title ''Modern algebra'', though a later, extensively revised edition in 1970 had the title ''Algebra''. The book was one of the first
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 ...
s to use an abstract axiomatic approach to
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
s,
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
s, and
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
s, and was by far the most successful, becoming the standard reference for graduate algebra for several decades. It "had a tremendous impact, and is widely considered to be the major text on algebra in the twentieth century." In 1975 van der Waerden described the sources he drew upon to write the book. In 1997
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...
recollected the book's influence: * Upon its publication it was soon clear that this was the way that algebra should be presented. * Its simple but austere style set the pattern for mathematical texts in other subjects, from
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
s to
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
theory. * an der Waerden'stwo volumes on modern algebra ... dramatically changed the way algebra is now taught by providing a decisive example of a clear and perspicacious presentation. It is, in my view, the most influential text of algebra of the twentieth century.


Publication history

''Moderne Algebra'' has a rather confusing publication history, because it went through many different editions, several of which were extensively rewritten with chapters and major topics added, deleted, or rearranged. In addition the new editions of first and second volumes were issued almost independently and at different times, and the numbering of the English editions does not correspond to the numbering of the German editions. In 1955 the title was changed from "Moderne Algebra" to "Algebra" following a suggestion of Brandt, with the result that the two volumes of the third German edition do not even have the same title. For volume 1, the first German edition was published in 1930, the second in 1937 (with the axiom of choice removed), the third in 1951 (with the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
reinstated, and with more on valuations). The fourth edition appeared in 1955 (with the title changed to ''Algebra''), the fifth in 1960, the sixth in 1964, the seventh in 1966, the eighth in 1971, the ninth in 1993. For volume 2, the first edition was published in 1931, the second in 1940, the third in 1955 (with the title changed to ''Algebra''), the fourth in 1959 (extensively rewritten, with elimination theory replaced by algebraic functions of 1 variable), the fifth in 1967, and the sixth in 1993. The German editions were all published by Springer. The first English edition was published in 1949–1950 and was a translation of the second German edition. There was a second edition in 1953, and a third edition under the new title Algebra in 1970 translated from the 7th German edition of volume 1 and the 5th German edition of volume 2. The three English editions were originally published by Ungar, though the 3rd English edition was later reprinted by Springer. * * * * * * There were also Russian editions published in 1976 and 1979, and Japanese editions published in 1959 and 1967–1971.


References

* * * * * * * * *{{Citation , last1=Taussky , first1=Olga , author1-link=Olga Taussky-Todd , title=Literaturberichte: Moderne Algebra , doi=10.1007/BF01708891 , year=1933 , journal=Monatshefte für Mathematik und Physik , issn=0026-9255 , volume=40 , issue=1 , pages=A3–A4, s2cid=124038027 History of mathematics Mathematics textbooks 1930 non-fiction books Algebra Springer Science+Business Media books