Moderne Algebra is a two-volume German textbook on graduate abstract algebra by Bartel Leendert van der Waerden (1930, 1931), originally based on lectures given by Emil Artin in 1926 and by Emmy Noether (1929) 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 textbooks to use an abstract axiomatic approach to groups, rings, and fields, 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"
There is a detailed discussion of the sources used for writing the book in van der Waerden (1975).
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 reinstated, and with more on valuations), the fourth 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.
- Hahn, H.; Taussky, Olga (1932), "Literaturberichte: Moderne Algebra", Monatshefte für Mathematik und Physik 39 (1): A11––A12, doi:10.1007/BF01699101, ISSN 0026-9255
- Hofreiter (1936), "Literaturberichte: Moderne Algebra", Monatshefte für Mathematik und Physik 45 (1): A25––A26, doi:10.1007/BF01708057, ISSN 0026-9255
- Hofreiter (1941), "Literaturberichte: Moderne Algebra", Monatshefte für Mathematik und Physik 49 (1): A21, doi:10.1007/BF01707351, ISSN 0026-9255
- Mac Lane, Saunders (1997), "Van der Waerden's Modern algebra", Notices of the American Mathematical Society 44 (3): 321–322, ISSN 0002-9920
- Ore, Oystein (1931), "Recent Publications: Reviews: Moderne Algebra", The American Mathematical Monthly 38 (4): 226, doi:10.2307/2301771, ISSN 0002-9890
- Ore, Oystein (1932), "Book Review: Moderne Algebra", Bulletin of the American Mathematical Society 38 (5): 327–329, doi:10.1090/S0002-9904-1932-05385-X, ISSN 0002-9904
- Ore, Oystein (1938), "Book Review: Moderne Algebra", Bulletin of the American Mathematical Society 44 (5): 320, doi:10.1090/S0002-9904-1938-06737-7, ISSN 0002-9904
- Noether, Emmy (1929), "Hyperkomplexe Größen und Darstellungstheorie", Mathematische Zeitschrift (Springer Berlin / Heidelberg) 30: 641–692, doi:10.1007/BF01187794, ISSN 0025-5874
- Schlote, K.-H. (2005), "Chapter 70 B. L. van der Waerden, Moderne Algebra", in Grattan-Guinness, Ivor, Landmark writings in western mathematics 1640--1940, Elsevier B. V., Amsterdam, pp. 901–916, ISBN 978-0-444-50871-3, MR 2169816
- Taussky, Olga (1933), "Literaturberichte: Moderne Algebra", Monatshefte für Mathematik und Physik 40 (1): A3––A4, doi:10.1007/BF01708891, ISSN 0026-9255
- van der Waerden, Bartel Leendert (1930), Moderne Algebra. Teil I, Die Grundlehren der mathematischen Wissenschaften 33, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56799-8, MR 0009016MR 0037277MR 0069787MR 0122834MR 0177027MR 0263581
- van der Waerden, Bartel Leendert (1931), Moderne Algebra. Teil II, Die Grundlehren der mathematischen Wissenschaften 34, Springer-Verlag, ISBN 978-3-540-56801-8, MR 0002841MR 0009016MR 0072829MR 0122835MR 0233647
- van der Waerden, Bartel Leendert (1949), Modern Algebra, New York, N. Y.: Frederick Ungar Publishing Co.
- van der Waerden, Bartel Leendert (1970), Algebra. Vol 1, Translated by Fred Blum and John R. Schulenberger, New York, N. Y.: Frederick Ungar Publishing Co., ISBN 978-0-387-40624-4, MR 0263582 Reprinted by Springer
- van der Waerden, Bartel Leendert (1970), Algebra. Vol. 2, Translated by John R. Schulenberger (3rd ed.), New York, N. Y.: Frederick Ungar Publishing Co., MR 0263583 Reprinted by Springer
- van der Waerden, Bartel Leendert (1975), "On the sources of my book Moderne Algebra", Historia Mathematica 2: 31–40, doi:10.1016/0315-0860(75)90034-8, ISSN 0315-0860, MR 0465721
- Zelinsky, Daniel (1951), "Book Review: Moderne Algebra", Bulletin of the American Mathematical Society 57 (3): 206, doi:10.1090/S0002-9904-1951-09496-3, ISSN 0002-9904