Basic Number Theory [Paperback]

From the reviews: L.R. Shafarevich showed me the first edition [&] and said that this book will be from now on the book about class field theory. In fact it is by far the most complete treatment of the main theorems of algebraic number theory, including function fields over finite constant fields, that appeared in book form. Zentralblatt MATH

)tPI(}jlOV, e~oxov (10CPUljlr1.'CWV Aiux., llpop. . .dsup.. The first part of this volume is based on a course taught at Princeton University in 1961-62; at that time, an excellent set of notes was prepared by David Cantor, and it was originally my intention to make these notes available to the mathematical public with only quite minor changes. Then, among some old papers of mine, I accidentally came across a long-forgotten manuscript by Chevalley, of pre-war vintage (forgotten, that is to say, both by me and by its author) which, to my taste at least, seemed to have aged very well. It contained a brief but essentially com? plete account of the main features of classfield theory, both local and global; and it soon became obvious that the usefulness of the intended volume would be greatly enhanced if I included such a treatment of this topic. It had to be expanded, in accordance with my own plans, but its outline could be preserved without much change. In fact, I have adhered to it rather closely at some critical points.I. Elementary Theory.- I. Locally compact fields.- ? 1. Finite fields.- ? 2. The module in a locally compact field.- ? 3. Classification of locally compact fields.- ? 4. Structure of p-fields.- II. Lattices and duality over local fields.- ? 1. Norms.- ? 2. Lattices.- ? 3. Multiplicative structure of local fields.- ? 4. Lattices over R.- ? 5. Duality over local fields.- III. Places of A-fields.- ? 1. A-fields and their completions.- ? 2. Tensor-products of commutative fields.- ? 3. Traces and norms.- ? 4. Tensor-products of A-fields and local fields.- IV. Adeles.- ? 1. Adeles of A-fields.- ? 2.l³"