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**A brief guide to algebraic number theory.**
*(English)*
Zbl 0963.11001

London Mathematical Society Student Texts. 50. Cambridge: Cambridge University Press. ix, 146 p. (2001).

The book contains an introduction to the theory of algebraic numbers. It contains 5 chapters. The first presents the principal theorems of that theory (using the ideal-theoretic approach): unique ideal factorization, Dirichlet’s unit theorem and the finiteness of the class number. In the case of a normal extensions the first three Hilbert groups are introduced. The second chapter introduces valuations and completions and uses them to prove the different theorem as well as the discriminant theorem. It concludes with the definition and principal properties of idéles and adèles. The theory is illustrated in the next chapter by quadratic, pure cubic, biquadratic and cyclotomic fields. As an application a proof of Fermat’s Last Theorem for regular prime exponents is given (using certain facts from class-field theory, which is presented without proofs in Chapter 5). Chapter 4 is devoted to analytic methods: Dedekind’s zeta-function and Hecke’s \(L\)-functions are introduced and the functional equation for them is established, following the steps of Tate’s thesis. The presentation of class-field theory in the last chapter is done both in the classical and modern way and is applied to give proofs of the quadratic reciprocity law for algebraic number fields and of the Kronecker-Weber theorem. There are 8 appendices explaining various algebraic and analytical topics used in the main body of the book.

This book seems to be ideally suited for non-specialists who would like to know quickly what algebraic number theory is about but who do not wish to study thick volumes dealing with that subject.

This book seems to be ideally suited for non-specialists who would like to know quickly what algebraic number theory is about but who do not wish to study thick volumes dealing with that subject.

Reviewer: Władysław Narkiewicz (Wrocław)