zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Algebraic number theory. Transl. from the German by Norbert Schappacher. (English) Zbl 0956.11021
Grundlehren der Mathematischen Wissenschaften. 322. Berlin: Springer. xvii, 571 p. DM 179.00; öS 1307.00; sFr. 162.00; £69.00; $ 99.00 (1999).

This is an English translation of a book, whose German original has been reviewed (Zbl 0747.11001). It brings in seven chapters a well-written introduction into modern number theory.

The first chapter presents the fundamental results including ideal theory (with Kummer’s factorization theorem), Dirichlet’s unit theorem, finiteness of the class-number and Hilbert’s ramification theory. One finds here also a study of orders, which is welcome, as this topic is usually omitted in most textbooks. Also a link to algebraic geometry is provided: one-dimensional schemes are defined, as well as the Picard and Chow groups and a connection with the theory of function fields is sketched.

The second chapter brings valuation theory, including a study of Henselian fields and their extensions and in the next chapter this is applied to algebraic number fields. The theory of the discriminant and different is presented, Arakelov ideals and Arakelov class group are considered and a proof is given of an analogue of the Riemann-Roch theorem, based on A. Weil’s definition of the genus for algebraic number fields. Then metrized modules over rings of integers are introduced and a formalism, which was introduced by A. Grothendieck in the case of algebraic varieties, is developed for these modules. After defining compactified Grothendieck groups (on which the tensor product induces a ring structure), the Chern character and Todd classes this construction culminates in the Grothendieck-Riemann-Roch theorem, relating, for a finite extension L/K of algebraic number fields, the Grothendieck groups corresponding to K and L. As the author states, this result “integrates completely the theory of algebraic integers into a general programme of algebraic geometry as a special case.”

The next two chapters present local and global class field theory, modelled upon a previous work of the author [“Class Field Theory”, Springer Verlag (1986; Zbl 0587.12001)] but including certain modifications and fresh examples.

The last chapter deals with zeta functions and L series. Contrary to most treatments of this topic the author does not use the approach based on harmonic analysis, but proceeds with a careful presentation of ideas of E. Hecke. This seems to be the first modern exposition of Hecke’s method.

This book is a most welcome addition to the literature and will serve as a learning tool for years to come. The translator made a splendid job, preserving the lucid style of the original.


MSC:
11RxxAlgebraic number theory: global fields
11SxxAlgebraic number theory: local and p-adic fields
11-02Research monographs (number theory)
11-01Textbooks (number theory)