Van der Waerden’s Modern Algebra. (English) Zbl 1036.01007

The author starts with a short information on how the two-volume 1931 text Modern Algebra (Zbl 0002.00804 and JFM 57.0153.03) was written and published “using lectures by E. Artin and E. Noether”. It formulated clearly and succintly the conceptual and structural insights which Noether had expressed so forcefully, combined with the elegance and understanding with which Artin had lectured. Upon its publication it was soon clear that this was the way in which algebra must now be presented.
For comparison, a two-volume text in algebra by Otto Haupt (published in 1928) is found helpful but heavy-handed – indeed, pedantic. It was van der Waerden who understood the real thrust of abstract algebra and who presented it abstractly but without pedantry.
Further the following van der Waerden’s activity is shortly characterized: the papers “Zur algebraischen Geometrie I–XX” in the Mathematische Annalen”, the text Introduction to Algebraic geometry (1939; Zbl 0021.25001; JFM 65.1393.01) (which was only the start of a needed reform of algebraic geometry, inter alia, of intersection multiplicity). He also mentions one dramatic moment in the long process in which algebraic geometry was gradually and totally transformed; this moment was connected with van der Waerden’s mistake in accepting a Derwidué paper for publication – this paper was then demolished in Oscar Zariski’s review (MR 0042171, see also Zbl 0045.42003). In connection with van der Waerden’s 1985 book “A History of algebra from Al Kharisme to Emmy Noether” (1985; Zbl 0569.01001) the author of the paper remarks: “We are fortunate that her (i.e. E. Noether’s) imagination has been made accessible by van der Waerden”.


01A60 History of mathematics in the 20th century