Conversational Introduction to Algebraic Number Theory : Arithmetic Beyond Z

Includes index.

Includes bibliographical references.

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

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.