An application to inseparable extensions.- II. Splitting places in separable extensions.- 6. Fourier transforms and standard functions.- 3. Lattices over algebraic number-fields.- 3. Tensor-products of A-fields and local fields.- IV. Tensor-products of commutative fields.- 3. Multiplicative structure of local fields.- 4. Lattices and duality over local fields.- 1. Classification of locally compact fields.- 4. The module in a locally compact field.- 3. 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." Zentralblatt MATH, 823 So, there is absolutely no example which illustrates the rather abstract material and brings it nearer to the heart of the reader. To develop this basic number theory on 312 pages efforts a maximum of concentration on the main features. 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. 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.
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