You are here

Quadratic Number Fields

Franz Lemmermeyer
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
Benjamin Linowitz
, on
Franz Lemmermeyer’s Quadratic Number Fields belongs to a growing genre of undergraduate textbooks that aim to make subjects that have traditionally been considered ‘graduate level’ accessible to broader audiences. The quintessential such book is John Stillwell’s gem Naive Lie Theory, which focuses its attention on matrix Lie groups in order to give its readers the flavor of the general theory without the many technicalities that make the subject so difficult for beginners. As its name suggests, Quadratic Number Fields is an introduction to the algebraic number theory of quadratic number fields. Quadratic number fields are much more concrete than the general number fields that are the usual subject of introductions to algebraic number theory, making them ideal for an undergraduate-level introduction to the subject, yet simultaneously have a theory that is rich and replete with interesting applications to classical number theoretic problems.  
The prerequisites of Quadratic Number Fields are minimal. All that is expected of the reader is a moderate background in linear algebra, abstract algebra (a standard year-long sequence covering groups, rings, and fields would suffice), and knowledge of elementary number theory through the theorem of quadratic reciprocity. In particular, the reader is not expected to have had any prior exposure to module theory. The lack of a module theory prerequisite greatly increases the book’s accessibility, as most introductory number theory texts covering standard topics like the structure of the unit group or the finiteness of the ideal class group either require that their readers have already seen modules, or else provide the reader with an appendix on module theory that is so terse as to be incomprehensible to anyone without prior exposure to the subject.
Quadratic Number Fields is a very historically motivated introduction to the algebraic number theory of quadratic fields. On its own, this statement may not appear to distinguish the text from the many other introductory algebraic number theory texts. After all, many, if not most, algebraic number theory textbooks are at least in part historically motivated. The typical structure of such a book is to begin with a chapter on Fermat’s Last Theorem which builds towards Kummer’s connection of potential solutions to Fermat’s equation to the arithmetic of certain number rings. After this, the historical motivation largely disappears, with the possible exception of brief digressions on the lives of Euler and Gauss. Overall, the reader is left with the impression that algebraic number theory is a field whose history ended sometime in the late 19th century. Lemmermeyer’s entire text, on the other hand, uses history to show how algebraic number theory has evolved from past to present and never really stops providing its readers with historical context. In this sense Quadratic Number Fields is quite similar to Oyestein Ore’s Number Theory and its History, a wonderful introduction to elementary number theory through the field’s history. The following example, taken at random from the text, will give readers an idea of what it is like to read Quadratic Number Fields. Chapter 7 of the text concerns the Pell Equation, a standard topic. After explaining the history of the equation, Lemmermeyer discusses its solvability. So far this is all relatively standard, though Lemmermeyer’s history of the equation is much richer and more detailed than the usual account. Next comes an old result of Harold Davenport, which is used to give Ankeny, Chowla, and Hasse’s 1965 construction of quadratic fields with nontrivial class number. On the following page, a 1990 theorem of Halter-Koch on lower bounds for class number is proven. In light of this example, it shouldn’t be surprising that Lemmermeyer’s short text has 136 references, the majority of which are to primary historical sources.
Although Quadratic Number Fields is historically motivated, it is also refreshingly modern. A great deal of modern algebraic number theory makes extensive use of arithmetic geometry. Apart from brief discussions of elliptic curves, algebraic number theory textbooks tend not to go into much depth about connections to geometry. Lemmermeyer, on the other hand, uses Diophantine equations extensively in order to provide concrete applications of the general theory (and vice versa). Examples include an interesting discussion of the arithmetic of Pell conics in order to motivate the elliptic curve group law, a chapter on Catalan’s Equation that begins with an 1850 result of Lebesgue and ends with a description of Preda Mihăilescu’s proof of the Catalan conjecture, and a proof of the law of quadratic reciprocity that makes extensive use of results on the number of rational points on an affine variety over a finite field.
To summarize: this is a wonderful, well-written introduction to modern algebraic number theory that has been made accessible to a broad undergraduate audience through the author’s restriction to quadratic number fields. Historically motivated, it shows how algebraic number theory has evolved over time and depicts it as living and breathing, not as a field that became static in the late 19th century.
Benjamin Linowitz ( is an Associate Professor of Mathematics at Oberlin College.