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**A history of abstract algebra. From algebraic equations to modern algebra.**
*(English)*
Zbl 1411.01005

Springer Undergraduate Mathematics Series. Cham: Springer (ISBN 978-3-319-94772-3/hbk; 978-3-319-94773-0/ebook). xxiv, 415 p. (2018).

“A history of abstract algebra” is a slightly ambiguous title. In the present case, history is understood in the traditional sense: How did modern abstract algebra develop, who are the main actors and what are the main ideas? Abstract algebra is indeed, as the name suggests, the set of ideas presented in B. L. van der Waerden’s classic book [Moderne Algebra. Bd. I. Berlin: Springer (1930; JFM 56.0138.01); Bd. II (1931; JFM 57.0153.03)], and it includes not only Galois theory, group theory and the theory of ideals, but also the beginnings of algebraic number theory and elementary algebraic geometry.

In some sense, the author begins where V. J. Katz and K. H. Parshall [Taming the unknown. A history of algebra from antiquity to the early twentieth century. Princeton, NJ: Princeton University Press (2014; Zbl 1304.01002)] stop. The intersection with B. L. van der Waerden’s [A history of algebra. Berlin etc.: Springer-Verlag (1985; Zbl 0569.01001)] is much more substantial. The starting point, however, is not, as one would perhaps expect, the solution of polynomial equations, but rather the theory of binary quadratic forms created by Lagrange and Gauss, including a discussion of cyclotomy and quadratic reciprocity. The unsolvability of the quintic, Galois theory and first steps in group theory by Jordan and Klein are covered next. The third thread is the development of algebraic number theory, which concentrates on the work of Dedekind, Kronecker and Hilbert. Finally, the beginnings of algebraic geometry (Brill-Noether, Lasker) are discussed; the last two lectures present the contributions of Emmy Noether and Van der Waerden’s book.

In very broad terms, the author presents the main strands in the evolution of abstract algebra; along the way he teaches the reader how to think as a historian of mathematics by emphasizing the importance of asking (the right) questions. He stresses repeatedly the necessity to consult the works of the masters and to check your sources – and in fact these sources, such as Gauss’s Disquisitiones, Jordan’s Traité, Dedekind’s Supplement, Kronecker’s Grundzüge and Hilbert’s Zahlbericht, play a more prominent role than in other works on the history of algebra. The author writes in an informal and colloquial style, making the book a pleasure to read.

The book contains a fair number of typos and minor errors. I will be content with mentioning two of them. On p. 280, the author claims that L. W. Reid, in his textbook [The elements of the theory of algebraic numbers. London: Macmillan and Co. (1910; JFM 41.0248.14)], denoted the rational numbers by \(k\) and the quadratic number field generated by \(i = \sqrt{-1}\) by \(k(i)\). Actually, Reid followed Hilbert in denoting the number field \(k\) generated (over the rationals) by the algebraic number \(\alpha\) by \(k(\alpha)\). For the second error the reviewer shares the responsibility: In Gauss’s fourth proof of the quadratic reciprocity law (pp. 331–332) based on the comparison of the quadratic Gauss sums \(\tau_p\), \(\tau_q\) and \(\tau_{pq}\) it is not sufficient to determine \(\tau_p\) only for prime values (this mistake already occurs in [F. Lemmermeyer, Reciprocity laws. From Euler to Eisenstein. Berlin: Springer (2000; Zbl 0949.11002)]).

The book under review is an excellent contribution to the history of abstract algebra and the beginnings of algebraic number theory. I recommend it to everyone interested in the history of mathematics.

In some sense, the author begins where V. J. Katz and K. H. Parshall [Taming the unknown. A history of algebra from antiquity to the early twentieth century. Princeton, NJ: Princeton University Press (2014; Zbl 1304.01002)] stop. The intersection with B. L. van der Waerden’s [A history of algebra. Berlin etc.: Springer-Verlag (1985; Zbl 0569.01001)] is much more substantial. The starting point, however, is not, as one would perhaps expect, the solution of polynomial equations, but rather the theory of binary quadratic forms created by Lagrange and Gauss, including a discussion of cyclotomy and quadratic reciprocity. The unsolvability of the quintic, Galois theory and first steps in group theory by Jordan and Klein are covered next. The third thread is the development of algebraic number theory, which concentrates on the work of Dedekind, Kronecker and Hilbert. Finally, the beginnings of algebraic geometry (Brill-Noether, Lasker) are discussed; the last two lectures present the contributions of Emmy Noether and Van der Waerden’s book.

In very broad terms, the author presents the main strands in the evolution of abstract algebra; along the way he teaches the reader how to think as a historian of mathematics by emphasizing the importance of asking (the right) questions. He stresses repeatedly the necessity to consult the works of the masters and to check your sources – and in fact these sources, such as Gauss’s Disquisitiones, Jordan’s Traité, Dedekind’s Supplement, Kronecker’s Grundzüge and Hilbert’s Zahlbericht, play a more prominent role than in other works on the history of algebra. The author writes in an informal and colloquial style, making the book a pleasure to read.

The book contains a fair number of typos and minor errors. I will be content with mentioning two of them. On p. 280, the author claims that L. W. Reid, in his textbook [The elements of the theory of algebraic numbers. London: Macmillan and Co. (1910; JFM 41.0248.14)], denoted the rational numbers by \(k\) and the quadratic number field generated by \(i = \sqrt{-1}\) by \(k(i)\). Actually, Reid followed Hilbert in denoting the number field \(k\) generated (over the rationals) by the algebraic number \(\alpha\) by \(k(\alpha)\). For the second error the reviewer shares the responsibility: In Gauss’s fourth proof of the quadratic reciprocity law (pp. 331–332) based on the comparison of the quadratic Gauss sums \(\tau_p\), \(\tau_q\) and \(\tau_{pq}\) it is not sufficient to determine \(\tau_p\) only for prime values (this mistake already occurs in [F. Lemmermeyer, Reciprocity laws. From Euler to Eisenstein. Berlin: Springer (2000; Zbl 0949.11002)]).

The book under review is an excellent contribution to the history of abstract algebra and the beginnings of algebraic number theory. I recommend it to everyone interested in the history of mathematics.

Reviewer: Franz Lemmermeyer (Jagstzell)

### MSC:

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |

01A05 | General histories, source books |

12-03 | History of field theory |

11-03 | History of number theory |