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Contents: Preface. 1. Getting our feet wet. 2. Cast of characters. 3. Quadratic number fields: First steps. 4. Paradise lost —and found. 5. Euclidean quadratic fields. 6. Ideal theory for quadratic fields. 7. Prime ideals in quadratic number rings. 8. Units in quadratic number rings. 9. A touch of class. 10. Measuring the failure of unique factorization. 11. Euler's prime-producing polynomial and the criterion of Frobenius–Rabinowitsch. 12. Interlude: Lattice points. 13. Back to basics: Starting over with arbitrary number fields. 14. Integral bases: From theory to practice, and back. 15. Ideal theory in general number rings. 16. Finiteness of the class group and the arithmetic of Z. 17. Prime decomposition in general number rings. 18. Dirichlet;s unit theorem, I. 19. A case study: Units in Z[v[3]2] and the Diophantine equation X3-2Y3±1. 20. Dirichlet's unit theorem, II. 21. More Minkowski magic, with a cameo appearance by Hermite. 22. Dedekind's discriminant theorem. 23. The quadratic Gauss sum. 24. Ideal density in quadratic number fields. 25. Dirichlet's class number formula. 26. Three miraculous appearances of quadratic class numbers. Index.

Gauss famously referred to mathematics as the “queen of the sciences” and to number theory as the “queen of mathematics”. This book is an introduction to algebraic number theory, meaning the study of arithmetic in finite extensions of the rational number field Q. Originating in the work of Gauss, the foundations of modern algebraic number theory are due to Dirichlet, Dedekind, Kronecker, Kummer, and others. This book lays out basic results, including the three “fundamental theorems”: unique factorization of ideals, finiteness of the class number, and Dirichlet’s unit theorem. While these theorems are by now quite classical, both the text and the exercises allude frequently to more recent developments. In addition to traversing the main highways, the book reveals some remarkable vistas by exploring scenic side roads. Several topics appear that are not present in the usual introductory texts. One example is the inclusion of an extensive discussion of the theory of elasticity, which provides a precise way of measuring the failure of unique factorization. The book is based on the author’s notes from a course delivered at the University of Georgia; pains have been taken to preserve the conversational style of the original lectures.