Modern algebra

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Modern Algebra is an influential two-volume textbook on algebra by Bartel Leendert van der Waerden , first published by Julius Springer in 1930/31 . It is based on lectures by Emmy Noether in Göttingen and Emil Artin in Hamburg , which van der Waerden - he was only 27 years old when the book was published and had come to Göttingen in 1924 - attended. It is considered the first modern textbook on algebra, based on the abstract, axiomatic and structured approach of the Hilbert- Noether School in Göttingen, which Richard Dedekind began at the end of the 19th century. This clearly distinguishes it from older textbooks on algebra, such as that of Heinrich Weber in particular , in which the theory of equations still played a major role, marks a turning point in the teaching of algebra and was a standard textbook for several decades.

Bulk

As one of the sources of the book, Van der Waerden gives: 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 and hypercomplex numbers (winter semester 1924/25 and winter semester 1927/28 in Göttingen). Emmy Noether, head of the Algebraic School in Göttingen in the 1920s, did not publish an algebra textbook himself, and Emil Artin only published much later. Artin originally wanted to write a textbook on algebra and hired van der Waerden, who also presented him with the first and second chapters and waited for the chapters Artin had promised. Soon after, however, Artin gave up on his intention. Van der Waerden worked alone on the book when he was a professor in Groningen (from 1927). In between he was a visiting professor in Göttingen in 1929, where he got married. In Groningen he was also in constant contact with Emmy Noether, who urged him to finish the book, even during his honeymoon. Emmy Noether herself had little pedagogical skills towards beginners and was mainly interested in research, and her lectures had the character of research seminars (later in exile at Bryn Mawr College , she used van der Waerden's textbook for beginners' lectures ).

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

In the USA, too, where, according to Garrett Birkhoff, algebra had played a subordinate role compared to analysis when the first edition was published, the book had a major influence on the establishment of algebra as a major factor, especially among younger mathematicians active research area. Another advantage was that the book was written in relatively simple German and was therefore also suitable for learning German. The first textbook of modern algebra in the USA was before the English translation of van der Waerden's book the Survey of Modern Algebra (1941) by Birkhoff and Saunders MacLane (the latter used van der Waerden's book in his algebra lectures at Harvard in 1935 ). As the first coherent representation of the then leading German algebraic school of abstract algebra (including mathematicians such as Helmut Hasse , Max Deuring , Wolfgang Krull , Richard Brauer , Otto Schreier and Ernst Steinitz , whose treatment of body theory from 1910 was of great influence for modern algebra was, belonged) van der Waerden's book also exerted a great influence on Nicolas Bourbaki in France. One of Bourbaki's lead authors, Jean Dieudonné , even wrote that Bourbaki's elements at the beginning of van der Waerden's book followed as a model. The mathematicians group Bourbaki developed the concept of mathematics as a theory of structures after the war.

Reinhard Siegmund-Schultze dealt with the reception history of the Noether School in the USA, in which Emmy Noether's concept of abstract algebra gradually supplanted the older approaches (represented in the USA by Leonard Dickson ), with van der Waerden's book being one played a significant role. The new abstract direction of the Noether school was later identified as "German" algebra, although there were other important algebraic schools in Germany, such as that of Issai Schur in Berlin, which dealt primarily with representation theory. There were also plans in 1935 (also supported by Emmy Noether) for a textbook on algebra from the Schur school, written by Richard Brauer , who had meanwhile emigrated to Toronto , which should also appear in Springer's basic teaching series, but which then ended up being broken. Not least on the advice of van der Waerden and Friedrich Karl Schmidt (both of whom were editors of the basic teaching along with Courant), Springer refrained from doing so. Even Richard Courant , who proposed the book by brewers as a textbook for students who would be put off by the abstract axiomatic thinking, advised, after he had long fought to the end of the publication by Springer in Germany since, as the connection to emigrants would make the publisher vulnerable to attacks by the National Socialists and there would already be enough algebra textbooks in Germany. Siegmund-Schultze sees this as a paradoxical triumph of the abstract direction of algebra (represented by van der Waerden's textbook), actually rejected by the National Socialists (for example Ludwig Bieberbach ), over more traditional approaches as a result of the break-up of the Schur school. However, that in no way meant that Van der Waerden's textbook on algebra for beginners had already established itself in Germany in the 1930s.

There have been some changes in the text over the years. In the second edition, van der Waerden removed the sections on well-order and transfinite induction - they were taken up again in the third edition - and avoided set-theoretical concepts ( axiom of choice , well-order theorem ) in body theory due to the ongoing discussions about the fundamentals of mathematics ( Brouwer and intuitionism). In the foreword, however, he expressed his regret that a completely finitistic approach, avoiding all non-constructive proofs of existence, would have been too great a sacrifice. Newer results such as valuation theory and in algebra theory (formerly hypercomplex numbers) were added in the 2nd and 3rd edition. In the 4th edition, the chapters on algebraic functions of a variable and topological algebra were added and the research of Nathan Jacobson (radical theory) and Wolfgang Krull (ideal theory) were incorporated. Van der Waerden had meanwhile worked intensively on the reconstruction of algebraic geometry, which was a source for many additions.

For the 7th edition in 1966, van der Waerden added a chapter on vector and tensor spaces, as the book, as he said in the foreword, was increasingly used as an introductory textbook for algebra as a whole, whereas originally he only wanted to give an introduction to modern algebra Basics of linear algebra (like determinant theory) assumed. A rearrangement of the two volumes, which were supposed to make the first volume a text for an introduction to algebra (with the exception of determinant theory), was already carried out in the second edition.

The textbook was reissued until 2003 (in the 1970s by Springer in the Heidelberger Taschenbücher series), but was increasingly replaced in teaching by other algebra textbooks that also use the category-theoretical approach, in particular the textbook by Serge Lang (Algebra , Addison-Wesley, first 1965).

In 1973 there was a controversy between Van der Waerden and Garrett Birkhoff (who was one of the early partisans of the Noether School in the USA) about the consideration of English ( Joseph Wedderburn on the theory of algebras) and American mathematicians (Dickson) by the Noether School and thus also in van der Waerden's textbook. According to Birkhoff, this was an example of how the Hilbert Courant School in Göttingen took over the results of other mathematicians. Van der Waerden replied that Wedderburn's contributions had a prominent place in Emmy Noether's lectures (though less so in Leonard Dickson's). That was also the reason for van der Waerden to publish an article on the sources of his book in the journal Historia Mathematica .

contents

Volume 1 in the 9th edition contains the chapters Numbers and Sets, Groups, Rings and Bodies, Vector Spaces and Tensor Spaces, Whole Rational Functions, Body Theory, Continuation of Group Theory, Galois' Theory, Order and Well-Ordering of Sets, Infinite Field Extensions, Real Bodies.

Volume 2 in its 9th edition contains the chapters Linear Algebra, Algebras, Representation Theory of Groups and Algebras, General Ideal Theory of Commutative Rings, Theory of Polynomial Ideals, Whole Algebraic Quantities, Weighted Fields, Algebraic Functions of a Variable, Topological Algebra (Chapter 20).

expenditure

Quotes

I am often asked for advice on how to start out studying algebra and to most people I say: First read Van der Waerden, in spite of what has been done since . Jean Dieudonné, 1970

literature

  • Review of the first edition by Øystein Ore , Bulletin AMS, Volume 38, 1932, pp. 327-329, online
  • Review of the second edition of Ore, Bulletin AMS, Volume 44, 1938, p. 320, online
  • Review of the third German edition and the first English edition by Daniel Zelinsky, Bulletin AMS, Volume 57, 1951, p. 206, online
  • W. Thimm, discussion of the 2nd edition (volume 1) in the annual report DMV, volume 49, 1939, p. 81, online
  • Gottfried Köthe , discussion of the 2nd edition (volume 2) in the annual report DMV, volume 51, p. 74, online
  • Heinrich Brandt, Review of the third edition (Volume 1), Annual Report DMV, Volume 55, 1952, pp. 47-48, online
  • Karl-Heinz Schlote B. L. van der Waerden, Modern Algebra in Ivor Grattan-Guinness Landmark writings in western mathematics 1640-1940 , Elsevier 2005
  • BL van der Waerden On the sources of my book Moderne Algebra , Historia Mathematica, Volume 2, 1975, pp. 31-40
  • Interview by Van der Waerden with Yvonne Dold-Samplonius , Notices AMS, March 1997, online
  • Saunders MacLane Van der Waerden's Modern Algebra , Notices AMS, March 1997, online
  • Paul Halmos : Some books of Auld Lang Syne , in P. Duren: A Century of Mathematics in America, Volume 1, AMS 1988, pp. 131-174
  • Reinhard Siegmund-Schultze: Mathematicians fleeing from Nazi-Germany, Princeton University Press 2009
  • Reinhard Siegmund-Schultze: Bartel Leendert van der Waerden (1903-1996) in the Third Reich: Modern Algebra in the Service of Anti-Modernism?, In: Dieter Hoffmann, Mark Walker (ed.), Foreign scientists in the Third Reich: the Debye- Affair in context, Göttingen: Wallstein, 2011, pp. 200–229
  • Mechthild Koreuber: Emmy Noether, the Noether School and Modern Algebra. On the history of a cultural movement , Springer Spectrum 2015, p. 232ff

Individual evidence

  1. ↑ Weber Textbook of Algebra , 3 volumes, Vieweg 1895-1908. Other well-known algebra textbooks were published in the 1920s by Helmut Hasse (Höhere Algebra 1926), Leonard Eugene Dickson (Modern algebraic theories, 1926, German 1929), Otto Haupt (Introduction to Algebra, 1929), Oskar Perron (Algebra 1927) .
  2. ^ Algebra, Volume 1, Introduction, Springer Verlag 2003, p. X
  3. ↑ published by Emmy Noether, Mathematische Zeitschrift, Volume 30, 1929, pp. 641-692. This publication was based on a transcript by van der Waerden, who also incorporated its content into the second volume of his Modern Algebra. Both lectures, three years apart, were on the same topic, representation theory of groups and algebras (called hypercomplex numbers) and each brought up the latest research by Noether. Van der Waerden, Meine Göttinger Lehrjahre , Mitteilungen DMV, Volume 5, 1997, Issue 2, p. 23, doi : 10.1515 / dmvm-1997-0208
  4. ↑ Among other things, an introduction to Galois theory, which also influenced later editions of van der Waerden's textbook. Foreword by van der Waerden to the 7th edition of the 1st volume in 1966
  5. ↑ Interview with Yvonne Dold-Samplonius, Notices AMS, Volume 44, March 1997
  6. ^ Garrett Birkhoff Current trends in algebra , American Mathematical Monthly, 80, 1973, 760-782
  7. ↑ Reinhard Siegmund-Schulze: Mathematicians fleeing from Nazi-Germany, Princeton University Press 2009, p. 291, citing Halmos (1988)
  8. ↑ Bourbaki Elements d´Histoire des Mathematiques , Masson 1984, p. 77. The textbook by Van der Waerden, published in 1930, united these works for the first time [meaning the German algebraic school] in one overview, opened the view and served as Guideline for many subsequent researches in abstract algebra , Le traité de Van der Waerden, publié en 1930, a reuni pour le première fois ces travaux en un exposé d´ensemble, ouvrant la voie et survant de guide aux multiples recherches d´Algèbre abstraites ultérieures
  9. ↑ Dieudonne, The work of Nicolas Bourbaki, American Mathematical Monthly, Volume 77, 1970, p. 136. Dieudonné also notes the great impression the book made on him and others when it was published in 1930 (he was working on his dissertation in Berlin at the time ), especially since algebra training in France was very rudimentary at the time.
  10. ^ Siegmund-Schultze, Mathematicians fleeing from Nazi Germany, p. 312
  11. ↑ Van der Waerden comments on this in the foreword to the 2nd edition. Siegmund-Schultze, Mathematicians fleeing from Nazi Germany, p. 313, sees it as concessions to the teaching of the time in National Socialist Germany with a tendency to reject abstract methods.
  12. ^ Siegmund-Schultze, Mathematicians fleeing from Nazi Germany, p. 316. Birkhoff wanted to publish an essay in Historia Mathematica on this subject, but then refrained from doing so. However, he expressed himself in his contribution The Rise of Modern Algebra in J. Dalton Tarwater, John Thomas White, John David Miller, Men and Institutions of American Mathematics, Texas Tech Press, 1976. Siegmund-Schultze notes that this is an American as Birkhoff accused the Noether School of doing something they themselves often practiced later, when the USA had become the dominant nation in mathematics.
  13. ↑ After Artin and Schreier. In Hamburg he met Artin and Schreier and learned from lectures by Erich Hecke .
  14. ↑ According to Van der Waerden, Meine Göttinger Lehrjahre , Mitteilungen DMV, Volume 5, 1997, Issue 2, 20-27, doi : 10.1515 / dmvm-1997-0208 , the presentation of the evaluation theory was based on the work of Alexander Markowitsch Ostrowski , whom he personally from Göttingen knew
  15. ^ Dieudonné, The work of Nicolas Bourbaki, American Mathematical Monthly, Volume 77, 1970, p. 137