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The mathematics of Frobenius in context. A journey through 18th to 20th century mathematics. (English) Zbl 1281.01002

Sources and Studies in the History of Mathematics and Physical Sciences. New York, NY: Springer (ISBN 978-1-4614-6332-0/hbk; 978-1-4614-6333-7/ebook). xiii, 699 p. (2013).
“By Albert’s time, Frobenius’ work on matrix algebra had become an anonymous part of basic mathematics”, the author writes on p. 385. In fact, this was the fate of most of the results obtained by Frobenius (a fate he shared, to some degree, with Dedekind and Emmy Noether). In fact, an average graduate student might find it difficult to answer the question about the main accomplishments of Frobenius. Our average graduate student would find giving an answer less problematic if he would read the book under review, which not only discusses the problems that Frobenius worked on but also provides information on the mathematical background.
Part I (the first 50 pages) deals with biographical information on Frobenius, his teachers and his colleagues. Part II (Berlin-style linear algebra) deals with the work of Kronecker and Weierstrass on the classification of quadratic and bilinear forms and sketches the early development of the theory of determinants, the principal axes theorem, the theory of elementary divisors, and Kronecker’s “disciplinary ideals”: these are (of course) not ideals in the mathematical sense, but ideals of the Berlin mathematical school.
After these preparations, the author presents the mathematics of Frobenius: his work on the problem of Pfaff (Chapter 6) (see also the author’s [Arch. Hist. Exact Sci. 59, No. 4, 381–436 (2005; Zbl 1078.01013)]), the Cayley-Hermite problem and matrix algebra (Chapter 7), linear algebra and elementary divisors (Chapter 8), group theory and the Frobenius density theorem (Chapter 9), abelian functions (Chapter 10), and Frobenius’ generalized theory of theta functions (Chapter 11). Next in line are some problems the author already has written about: group determinants and group characters (Chapters 12, 13) (see [Arch. Hist. Exact Sci. 7, 142–170 (1970; Zbl 0217.29903); Arch. Hist. Exact Sci. 12, 217–243 (1974; Zbl 0397.01005)]), and representation theory (Chapters 14, 15) [Arch. Hist. Exact Sci. 8, 243–287 (1972; Zbl 0255.01005); Sémin. Congr. 3, 69–100 (1998; Zbl 1044.01526)]. The last two chapters deal with “loose ends” (problems in the theory of matrices) and “nonnegative matrices” and connections with continued fractions [Arch. Hist. Exact Sci. 62, No. 6, 655–717 (2008; Zbl 1171.01007)].
The author has succeeded admirably in describing the mathematical work of Frobenius. To say that this book is an excellent contribution to the mathematical literature is a huge understatement: it is, or should be, a role model for historical writing, and for bringing the mathematics of the recent past back to life.

MSC:

01-02 Research exposition (monographs, survey articles) pertaining to history and biography
01A70 Biographies, obituaries, personalia, bibliographies
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
11-03 History of number theory
15-03 History of linear algebra
20-03 History of group theory
30-03 History of functions of a complex variable

Biographic References:

Frobenius, Georg
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