##
**Number theory. English translation, edited and prepared for publication by Horst Günter Zimmer. 3rd ed., corr. and enl.**
*(English)*
Zbl 0423.12001

Berlin: Akademie-Verlag. xvii, 638 p., 49 figs. M 88.00 (1979).

This translation of Hasse’s book is very timely since there are quite a few fundamental works on algebraic number theory like this. The main topic of the book is the foundation of number theory in algebraic number fields, in which a major role is played by the divisor-theoretic approach with the valuation theory developed by Hensel. This translation was prepared on the basis of the third German edition [“Zahlentheorie”, 3rd edition. Berlin: Akademie-Verlag (1969)]; there is only one major alteration in chapter 16 in comparison with the German edition.

The book consists of three parts: I. The foundations of arithmetic in the rational number field; II. The theory of valued field; III. The foundations of arithmetic in algebraic number field.

In part I the elementary number theory is discussed in a field \(K\) with an integral domain whose quotient field is \(K\). Among others the properties of congruence, residue classes, characters of finite Abelian groups, and the quadratic residues are detailed.

Part II is devoted to the treatment of the valuations and the valued field. It deals with archimedean, non-archimedean and discrete valuations of a field; the completion of a valued field; \(p\)-adic fields; the prolongation of the valuation to transcendental and algebraic extensions; the structure of a finite-algebraic extension of a complete valued field; furthermore with the divisors and units in a discrete valued field. The elements of arithmetic in a discrete valued field are also introduced.

In part III an arithmetic of the algebraic number fields is developed, using that some properties of the complete system of inequivalent valuations of rational number field carry over to the finite-algebraic extensions. Applying the previous concept and results to the treatment of quadratic number field, there are discussed decomposition law, discriminant, integral basis, and norm of a number. Other applications of the previous results are the discussion of cyclotomic field, the group of units of an algebraic number field, and the finiteness of the class number of an algebraic number field. As examples, the units and the class numbers are also detailed for quadratic and cyclotomic fields.

The foundation of number theory in algebraic function field with one indeterminate is also a topic of the book. At the end of most chapters one can find a treatment for function fields. The style of the book is concise. Only the main results are written as theorems, the other results are written in italics.

The book consists of three parts: I. The foundations of arithmetic in the rational number field; II. The theory of valued field; III. The foundations of arithmetic in algebraic number field.

In part I the elementary number theory is discussed in a field \(K\) with an integral domain whose quotient field is \(K\). Among others the properties of congruence, residue classes, characters of finite Abelian groups, and the quadratic residues are detailed.

Part II is devoted to the treatment of the valuations and the valued field. It deals with archimedean, non-archimedean and discrete valuations of a field; the completion of a valued field; \(p\)-adic fields; the prolongation of the valuation to transcendental and algebraic extensions; the structure of a finite-algebraic extension of a complete valued field; furthermore with the divisors and units in a discrete valued field. The elements of arithmetic in a discrete valued field are also introduced.

In part III an arithmetic of the algebraic number fields is developed, using that some properties of the complete system of inequivalent valuations of rational number field carry over to the finite-algebraic extensions. Applying the previous concept and results to the treatment of quadratic number field, there are discussed decomposition law, discriminant, integral basis, and norm of a number. Other applications of the previous results are the discussion of cyclotomic field, the group of units of an algebraic number field, and the finiteness of the class number of an algebraic number field. As examples, the units and the class numbers are also detailed for quadratic and cyclotomic fields.

The foundation of number theory in algebraic function field with one indeterminate is also a topic of the book. At the end of most chapters one can find a treatment for function fields. The style of the book is concise. Only the main results are written as theorems, the other results are written in italics.

Reviewer: Péter Kiss (Eger)

### MSC:

11Rxx | Algebraic number theory: global fields |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

11R58 | Arithmetic theory of algebraic function fields |

11Sxx | Algebraic number theory: local fields |

12J10 | Valued fields |

12J20 | General valuation theory for fields |

12J25 | Non-Archimedean valued fields |