From Natural Numbers to Quaternions

This attractive book is based on a clever idea: Kramer and von Pippich lead the reader through an introduction to the ideas of abstract algebra inspired by the construction of number systems. Beginning from Peano’s axioms as a characterization of the natural numbers, the authors develop the fundamentals of number theory. Next, properties of natural numbers are abstracted to motivate the definitions of semigroup and group, and the authors drive rapidly toward quotient groups. We learn how every regular semigroup can be embedded in a group, and as an example of the process, we construct the integers. This process is immediately repeated with rings and quotient rings to construct the rationals. The notion of decimal expansion now leads to rational Cauchy sequences, and thus to the real numbers. At this point, the authors boldly write, “We define \(i:=\sqrt{-1}\)” and proceed rapidly to prove the fundamental theorem of algebra and the transcendence of \(e\). The book concludes with a description of the quaternion algebra and the group structure it induces on the \(3\)-sphere.

Each chapter ends with an appendix intended to introduce readers to active research topics. I found it somewhat jarring, at the end of the first chapter, to pass in a few pages from the division algorithm to the functional equation \[\pi^{-s/2}\Gamma\left(\frac s2\right)\zeta(s) = \pi^{-(1-s)/2}\Gamma\left(\frac{1-s}{2}\right)\zeta(1-s)\] satisfied by the (meromorphic continuation to the complex plane of the) Riemann zeta function, even if a connection exists between the two topics via prime numbers. I sympathized with the desire to give the readers an idea of what the Riemann hypothesis says, but suspected that few members of the target audience could absorb it, and feared that readers might find it discouraging, even in an appendix. The same could be said of the Birch-Swinnerton-Dyer conjecture, which appears in Appendix C. Other appendices, addressing topics like octonions and the \(p\)-adics, seem more in line with the rest of the text, both in level and in focus.

A number of exercises are interspersed with the text. Relatively few of these are routine. Many are broad, such as

II.4.15(c)  “Determine all possible groups of orders 4 and 6 up to isomorphism”

or open-ended, such as

II.1.14(b)  “Find other examples of semigroups that are not monoids”

and, more tantalizingly,

III.4.4  “Try to find a skew field with finitely many elements that is not a field”.

Solutions to most of the exercises, and hints to some of the others, appear in the back of the book.

The authors list only a few prerequisites for the reader: naive set theory, injective and surjective mappings, and (in later chapters only) finite-dimensional vector spaces and some ideas from calculus. This assessment is fair if we leave aside the appendices, which, after all, are not meant to be read as complete self-contained rigorous treatments of their subjects. The text does, however, expect the reader to be fairly mature mathematically, and a wise instructor will supplement it liberally with examples.

From Natural Numbers to Quaternions is emphatically not a traditional abstract algebra textbook. You will not find the definition of the center of a group here. Instead, the authors have maintained their focus on the construction of number systems, and they have used that framework to introduce readers to the ideas of algebra and to the outlines of some famous results and challenges related to ongoing research. They have done their work well. It will be of particular interest to anyone who is just embarking on a career in mathematics.

James A. Swenson is a professor of mathematics at the University of Wisconsin-Platteville, where he spends most of his time teaching future engineers. He earned the Ph.D. at the University of Minnesota (Twin Cities), studying algebraic topology under the direction of Mark Feshbach. In his spare time, he sings in one choir and directs another.