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**Hurwitz’s lectures on the number theory of quaternions.**
*(English)*
Zbl 1526.01002

Heritage of European Mathematics. Berlin: European Mathematical Society (EMS) (ISBN 978-3-98547-011-2/hbk; 978-3-98547-511-7/ebook). xvii, 293 p. (2023).

In his lectures on the arithmetic of the quaternions [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1896, 314–340 (1896; JFM 27.0162.01); Vorlesungen über die Zahlentheorie der Quaternionen. Berlin: J. Springer (1919; JFM 47.0106.01)], A. Hurwitz developed a theory of quaternions in parallel to the arithmetic of Gaussian integers: he adjoins the elements \(i\), \(j\), \(k\) and \(\frac{1+i+j+k}{2}\) to the integers and shows that the resulting ring admits a Euclidean algorithm, and that it has unique factorization. From this he deduces Lagrange’s theorem that every natural number is the sum of at most four squares, and discusses Euler’s problem of magic squares of squares. The authors have now translated these lectures and have provided the readers with detailed historical comments.

In addition, the authors discuss the differences between the arithmetic of Lipschitz and Hurwitz quaternions, they explain entries in Hurwitz’s mathematical diaries dealing with quaternions, present an English translation of Hurwitz’s article on the composition of quadratic forms [Math. Ann. 88, 1–25 (1922; JFM 48.1164.03)], and briefly go into generalizations of the quaternions, in particular octonions.

There are a few articles that are directly related to Hurwitz’s work and perhaps should have been mentioned; they concern the connection between quaternions and the arithmetic of binary quadratic forms. B. Venkov [Leningrad, Bull. Ac. Sc. (6) 16, 205–220 (1922; JFM 50.0077.01)] gave a beautiful proof of a deep result in Gauss’s Disquisitiones arithmeticae connecting the number of ways in which \(m\) can be written as a sum of three squares and the class number of \(\mathbb{Q}(\sqrt{-m})\). A modernized proof was given by H. P. Rehm [Lect. Notes Pure Appl. Math. 79, 31–38 (1982; Zbl 0499.10020)] (see also T. R. Shemanske [J. Number Theory 23, 203–209 (1986; Zbl 0585.10012)] and, in particular, P. Hanlon’s thesis [Applications of the quaternions to the study of imaginary quadratic ring class groups. Pasadena, CA: California Institute of Technology (PhD Thesis) (1981)]). We finally mention that B. Rice [Proc. Am. Math. Soc. 27, 1–7 (1971; Zbl 0212.06601)] showed how to derive the composition of binary quadratic forms using quaternions.

Hurwitz’s lectures on the arithmetic of quaternions can be read with profit by undergraduates familiar with elementary number theory. The editors did a very fine job of making these lectures accessible to today’s readers.

In addition, the authors discuss the differences between the arithmetic of Lipschitz and Hurwitz quaternions, they explain entries in Hurwitz’s mathematical diaries dealing with quaternions, present an English translation of Hurwitz’s article on the composition of quadratic forms [Math. Ann. 88, 1–25 (1922; JFM 48.1164.03)], and briefly go into generalizations of the quaternions, in particular octonions.

There are a few articles that are directly related to Hurwitz’s work and perhaps should have been mentioned; they concern the connection between quaternions and the arithmetic of binary quadratic forms. B. Venkov [Leningrad, Bull. Ac. Sc. (6) 16, 205–220 (1922; JFM 50.0077.01)] gave a beautiful proof of a deep result in Gauss’s Disquisitiones arithmeticae connecting the number of ways in which \(m\) can be written as a sum of three squares and the class number of \(\mathbb{Q}(\sqrt{-m})\). A modernized proof was given by H. P. Rehm [Lect. Notes Pure Appl. Math. 79, 31–38 (1982; Zbl 0499.10020)] (see also T. R. Shemanske [J. Number Theory 23, 203–209 (1986; Zbl 0585.10012)] and, in particular, P. Hanlon’s thesis [Applications of the quaternions to the study of imaginary quadratic ring class groups. Pasadena, CA: California Institute of Technology (PhD Thesis) (1981)]). We finally mention that B. Rice [Proc. Am. Math. Soc. 27, 1–7 (1971; Zbl 0212.06601)] showed how to derive the composition of binary quadratic forms using quaternions.

Hurwitz’s lectures on the arithmetic of quaternions can be read with profit by undergraduates familiar with elementary number theory. The editors did a very fine job of making these lectures accessible to today’s readers.

Reviewer: Franz Lemmermeyer (Jagstzell)

### MSC:

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

11-03 | History of number theory |

01A55 | History of mathematics in the 19th century |

01A75 | Collected or selected works; reprintings or translations of classics |

11E04 | Quadratic forms over general fields |

11R52 | Quaternion and other division algebras: arithmetic, zeta functions |