The Classical Fields: Structural Features of the Real and Rational Numbers

One thing I always want to make sure that our mathematics majors understand upon graduating is the important rings (the integers, the rational numbers, the reals, and the complex numbers) and how they are constructed. This book takes that idea to a whole new level. The authors present a study of the classical fields (the last three of the rings listed above) from several different aspects.

The first chapter presents the field of real numbers. This set is examined in the context of its additive group and multiplicative group, as an ordered set, a topological space, a measure space, a field, and then a topological field. The second chapter uses ultraproducts to construct the nonstandard reals from the rationals. The properties of the nonstandard numbers can then be used to deduce properties for the reals.

The third chapter concerns the rational numbers (as a subfield of the reals): the addition and multiplication, the ordering, the additive and multiplicative topologies, and the p-adic rationals. The fourth chapter uses the notion of completions to construct the reals from the rationals. The fifth chapter covers the construction of the p-adic rationals and integers. Finally, there is an appendix with a collection of facts about ordinal and cardinal numbers, topological group theory, and field theory.

There are some exercises scattered throughout the text, which might make it tempting to use in a classroom. But it’s such a specialized topic; maybe half of the book would be accessible to a good graduate student. Which is, in some sense, a pity, since there is some nice material that graduate and even undergraduate students should be exposed to here.

Donald L. Vestal is Assistant Professor of Mathematics at South Dakota State University. His interests include number theory, combinatorics, spending time with his family, and working on his hot sauce collection. He can be reached at Donald.Vestal(AT)sdstate.edu.