First the bad news. This book is a hodgepodge of results from abstract algebra, number theory, and control theory, with no obvious organizing principles. It is loaded with typographical errors, both in the English and the mathematics. The English itself is often shaky, with wrong words being used and a sometimes impenetrable syntax. My favorite is on p. 81 where we read that double integrals can be transformed into respected integrals. And what are we to make of examples such as this (p. 20)?

Since anyway, rings and fields are abstract number systems, we model on the latter and assume distributive law as we have in the number system. Commutative law for addition is naturally added as in the number system while that for multiplication is rarely found in nature.

Or “piervotnyi koren” on p. 10? Some web sleuthing suggests that this is a combination of Polish and Russian for primitive root, which is in fact the subject being discussed.

Despite the title, there are, as far as I can tell, no applications of number theory in the book. The last chapter deals with control theory, which applies some of the same methods from mathematical analysis that number theory uses, in particular several forms of the Fourier transform, but number theory itself is not used.

Now the good news: The book includes four proofs of quadratic reciprocity, all interesting and none found in commonly-used textbooks. The expositions in the book are concise and appear to be correct, once you get past the typos.

And that’s all the news for this book.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.