This book is written in the spirit of the early forties and just this makes it a valuable source of information for everyone who is working about problems related to number and function fields. Wiless proof uses sophisticated mathematical techniques that we will not be able to describe in detail, but in Chapter 30 we will prove that no fourth power is. So, there is absolutely no example which illustrates the rather abstract material and brings it nearer to the heart of the reader. There is nothing original to me in the notes. High school mathematics, familiarity with proofs by mathematical induction and with the basic properties of limits of sequences of real numbers (in particular. To develop this basic number theory on 312 pages efforts a maximum of concentration on the main features. These are the notes of the course MTH6128, Number Theory, which I taught at Queen Mary, University of London, in the spring semester of 2009. The spirit of the book is the idea that all this is asic number theory' about which elevates the edifice of the theory of automorphic forms and representations and other theories. The theory is presented in a uniform way starting with topological fields and Haar measure on related groups, and it includes not only class field theory but also the theory of simple algebras over local and global fields, which serves as a foundation for 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. To prove a statement P for all positive integers at least n0. Shafarevich showed me the first edition in autumn 1967 in Moscow and said that this book will be from now on the book about class field theory. Therefore, the identity is true for all natural numbers n. In fact, I have adhered to it rather closely at some critical points. It had to be expanded, in accordance with my own plans, but its outline could be preserved without much change. 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. Of course, knowledge and familiarity with basic number theory (for instance, the distribution of primes up to the Bombieri-Vinogradov Theorem) are helpful.
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. 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. )tPI(}jlOV, e~oxov (10CPUljlr1.'CWV Aiux., llpop.dsup.
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