Modern Algebra

Modern algebra is an influential two-volume textbook on algebra of Bartel Leendert van der Waerden, the first in 1930/31 was published by Julius Springer. It is based on lectures by Emmy Noether and Emil Artin in Göttingen in Hamburg, the van der Waerden - he was with the book was published only 27 years old and came to Göttingen in 1924 - attended. It is considered the first modern textbook on algebra, based on the abstract, axiomatic and structural emphasized access, the Hilbert -Noether school in Göttingen, which was begun by Richard Dedekind end of the 19th century. Thus, it differs significantly from older textbooks of algebra, in particular that of Heinrich Weber, in which even the theory of equations played a big role, marked a turning point in the teaching of algebra and was for several decades a standard textbook.

Bulk

As one of the sources of the book Van is der Waerden on: Algebra Lecture by Emil Artin (summer semester 1926 Hamburg), a seminar Ideal theory in Hamburg in the winter semester 1926/27, ( Emil Artin, Wilhelm Blaschke, Otto Schreier ), lectures by Emmy Noether on group theory (winter semester 1924/25 in Göttingen) and hyper complex numbers (winter semester 1927/28 in Göttingen). Emmy Noether, the head of the algebraists School in Göttingen in the 1920s, published even no algebra textbook and Emil Artin until much later. Originally Artin wanted to write a textbook of algebra and tensed van der Waerden for one, who presented him the first and second chapter, waiting for the promised Artin chapter. Soon after, however, Artin was his intention. Van der Waerden worked alone on the book when he (from 1927) was a professor in Groningen. In between, he was in 1929 as a visiting professor in Göttingen, where he married. He stood in Groningen also in constant contact with Emmy Noether, who urged him to complete the book.

The 4th edition in 1955 had only algebra in the title, a proposal by Heinrich Brandt in the discussion of the third edition in 1955 following.

Even in the U.S., where, in the words of Garrett Birkhoff Algebra in the appearance of the first edition by then compared to the Analysis played a minor role, the book exercised especially in younger mathematicians from the beginning of a major impact in the establishment of the algebra as active area of ​​research. The first textbook of modern algebra in the U.S. was before the English translation of van der Waerden 's book of the Survey of Modern Algebra (1941 ) by Birkhoff and Saunders MacLane. As a first coherent account of the then leading into the Algebra algebraic German school ( which included the addition Mentioned mathematicians such as Helmut Hasse, Max Deuring, Wolfgang Krull, Richard Brauer, Otto Schreier and Ernst Steinitz ) practiced van der Waerden 's book is also a large influence on Nicolas Bourbaki in France. The mathematician Bourbaki group developed the conception of mathematics as a theory of structures continue after the war.

There were some changes in the text over the years. In the second edition van remote der Waerden the sections on well-ordering and transfinite induction - they were taken up again in the third edition - and avoided set-theoretical concepts ( the axiom of choice, well-ordering theorem ) in the field theory due to the then ongoing discussions on the foundations of mathematics ( Brouwer and intuitionism ). However, in the preface he expressed regret that a fully finitistischer access avoiding all nichtkonstruktiver existence proofs would have been too great a sacrifice. Recent results, including the assessment theory and in Algebrentheorie (formerly hyper complex numbers ) were added in the 2nd and 3rd edition. In the 4th edition, the chapter came on Algebraic functions of one variable and Topological Algebra added and the research of Nathan Jacobson (radical theory ) and Wolfgang Krull (ideal theory ) were included. Van der Waerden himself had now dealt intensively with a reconstruction of algebraic geometry, which was a source of many supplements.

For the 7th edition 1966 van der Waerden added a chapter vector and Tensorräume, since the book as he said in the preface, has been increasingly used as a total introductory textbook for algebra, as he originally wanted to give an introduction to modern algebra and Basics of linear algebra (such as theory of determinants ) presupposed. A rearrangement of the two volumes, which should make the first band to a text for Introduction to Algebra (except determinants theory ), but he had already taken place in the second edition.

The textbook has been to 2003, re-issued ( in the 1970s by Springer in the series Heidelberger paperbacks ), but was in teaching increasingly from other algebra textbooks, which also use the category theoretical approach, displaced, especially the textbook by Serge Lang ( Algebra, Addison Wesley, first 1965).

Volume 1 contains the chapter numbers and quantities, groups, rings and fields, vector spaces and Tensorräume, Full Rational Functions, body theory, continuation of group theory, the theory of Galois, order and well-ordering of sets, infinite field extensions, Real body in the 9th edition.

Volume 2 contains the chapters linear algebra, algebra, representation theory of groups and algebras, General theory of ideals in commutative rings, Theory of Polynomial, whole algebraic sizes, rated body, algebraic functions of one variable, Topological Algebra (Chapter 20) in the 9th edition.

Expenditure

  • Modern Algebra. Using the lectures by E. Artin and E. Noether, 1st edition, 2 volumes, Berlin: Springer Verlag Julius, 1930, 1931, (243 and 216 pages), basic teachings of the mathematical sciences 2nd edition, 1937, 1940
  • 3rd Edition, 1951, 1955
  • 4th Edition as Algebra, 2 volumes, 1955, 1959,
  • Volume 1 appeared in 1966 in 7th edition, 1971 in 8th edition and 1993 9th edition, volume 2 1993 6th Edition
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