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Quadratic Irrationals: An Introduction to Classical Number Theory

Franz Halter-Koch
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Pure and Applied Mathematics
[Reviewed by
Allen Stenger
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This is an interesting, discursive, and I think successful attempt to present a cohesive treatment of several important topics in classical number theory by first developing a theory of quadratic irrationals (irrational numbers that are zeroes of quadratic polynomials with integer coefficients) and then building on that to show how the other topics reflect different faces of this theory. The topics covered include Pell’s equation, quadratic forms, class groups, class numbers, and quadratic orders (“orders” in the sense of rings of algebraic integers).

The book attempts to be self-contained, and so covers a lot of needed number-theory topics that are not closely related to quadratic irrationals. This sometimes makes the thread hard to follow. There is a complete proof of Dirichlet’s theorem on primes in arithmetic progressions, a result that is used in several places. There are are chapters on continued fractions and on quadratic residues and Gauss sums. This supplemental material follows classical approaches, and so is similar to what is found in many general-purpose number theory books. There is also a chapter on cubic and quadratic reciprocity, which again is not tightly tied to quadratic irrationals but uses many of the techniques that were developed for these.

Although this is classical material, it is described in modern terms using abstract algebra. The level of expertise in algebra is moderate (mostly rings and structure theorems), and the book includes a substantial appendix summarizing the concepts and main results that will be used. There is a shorter but similar appendix on classical analysis.

The book is subtitled “An Introduction to Classical Number Theory”, and while its extensive background material makes this technically true, it seems unlikely that anyone wanting to be introduced to classical number theory would want to start here; the focus is too narrow. I believe it works best as a monograph for those who are already familiar with some parts of the material covered here and would like to see other approaches. The book is skimpy on exercises, which also argues against use as a textbook. On the plus side, it does include a better selection of numerical examples than is found in most books, and it has a number of applications to other areas such as diophantine equations. A good book that has a similar approach and a good bit of overlap in material is Ireland & Rosen’s A Classical Introduction to Modern Number Theory.

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, a math help site that fosters inquiry learning.

Quadratic Irrationals
Quadratic irrationals, quadratic number fields and discriminants
The modular group
Reduced quadratic irrationals
Two short tables of class numbers

Continued Fractions
General theory of continued fractions
Continued fractions of quadratic irrationals I: General theory
Continued fractions of quadratic irrationals II: Special types

Quadratic Residues and Gauss Sums
Elementary theory of power residues
Gauss and Jacobi sums
The quadratic reciprocity law
Sums of two squares
Kronecker and quadratic symbols

L-Series and Dirichlet’s Prime Number Theorem
Preliminaries and some elementary cases
Multiplicative functions
Dirichlet L-functions and proof of Dirichlet’s theorem
Summation of L-series

Quadratic Orders
Lattices and orders in quadratic number fields
Units in quadratic orders
Lattices and (invertible) fractional ideals in quadratic orders
Structure of ideals in quadratic orders
Class groups and class semigroups
Ambiguous ideals and ideal classes
An application: Some binary Diophantine equations
Prime ideals and multiplicative ideal theory
Class groups of quadratic orders

Binary Quadratic Forms
Elementary definitions and equivalence relations
Representation of integers
Theory of genera
Ternary quadratic forms
Sums of squares

Cubic and Biquadratic Residues
The cubic Jacobi symbol
The cubic reciprocity law
The biquadratic Jacobi symbol
The biquadratic reciprocity law
Rational biquadratic reciprocity laws
A biquadratic class group character and applications

Class Groups
The analytic class number formula
L-functions of quadratic orders
Ambiguous classes and classes of order divisibility by 4
Discriminants with cyclic 2-class group: Divisibility by 8 and 16

Appendix A: Review of Elementary Algebra and Number Theory
Appendix B: Some Results from Analysis



List of Symbols

Subject Index