This is an introductory text in algebraic number theory that has good coverage but is marred by many small errors and by a narrative that jumps back and forth and sometimes leaves gaps. This second edition is completely reorganized and rewritten from the first edition.
The book is slanted strongly toward the “algebraic” side rather than the “number theory” side, and in fact we don’t see any questions involving the rational integers until Chapter 4. The approach is to start with general structures, such as integral domains, and gradually add conditions until we are ready to apply the results to number fields. Happily, the book avoids extreme abstraction by introducing the quadratic fields and cyclotomic fields early and using these for frequent examples. Unhappily, there’s no mention of computers, despite the importance of programs such as PARI/GP in algebraic number theory.
Very Good Features: (1) The applications are not limited to Diophantine equations, as in many books, but cover a wide range, including factorization into primes, primality testing, and the higher reciprocity laws. (2) The book has a large number of mini-biographies (about half a page each) of the number theorists whose work is being discussed. No references are given for these biographies, but they appear to be accurate.
A novel feature is an appendix of Latin phrases with explanations. Peculiarly, no mathematical phrases are listed and only phrases used in non-technical English are included; for example reductio ad absurdum and quod erat demonstrandum are omitted.
The writing is often awkward. Some samples:
- Despite this, she was dismissed from her position at the University of Göttingen in 1933 due to the Nazi rise to power, given that she was Jewish. (p. 23)
- By Corollary 1.13 on page 37 (in view of the comment on condition (A) in Remark 1.12 on page 26), condition (A) of Definition 1.23 on page 25 is satisfied. (p. 42)
The explanations sometimes seem needlessly complicated. For example, in several places the book appeals to the well-ordering principle to conclude that a set of positive integers has a least element — a fact that most people (certainly number theorists) would consider obvious.
A disorienting feature of the book is that the proofs often refer forward to later pages for some results they need. This happens so often that it makes the book look like it was written out of order. In the book’s defense, most of these are references to problems at the end of the section, and to some extent are just an idiom for “proof left to the reader”. But the book refers forward even for examples; a real example would be worked out fully where it occurred. A more serious cause of disorientation occurs when the book uses terminology or symbols that have not been defined yet; for example on p. 19 it uses the norm notation NF that won’t be defined until p. 65; on p. 22 it talks about two algebraic numbers being relatively prime (this concept is later defined for ideals, but I think it is never defined for algebraic numbers); and on p. 43 it uses the conjugates of an algebraic number, which won’t be defined until p. 62.
There are many small errors. Some of these are misspellings and not likely to confuse. Others are more substantive, such as in Proposition 1.1 on p. 41 where the wrong base field is specified. This is precisely the kind of error that derails students, who won’t realize that the statement as written is absurd and will spend hours trying to prove or understand it. There are occasional gaps in the reasoning. For example, on p. 40 the book (essentially) defines the cyclotomic polynomial as the monic polynomial whose roots are the primitive nth roots of unity, but then assumes implicitly and without proof that the coefficients of this polynomial are rational numbers. In all cases that I checked, the stated result was correct even though its proof was incomplete.
Bottom line: a book with good coverage and a lot of promise, that could have been an excellent text if it had been constructed more carefully. Another book that has similar coverage is Marcus’s Number Fields. Marcus’s book is very concrete, is slanted much more toward the “number theory” side, and has excellent exercises; but lacks the breadth of applications in Mollin’s 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.