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**The Brauer-Hasse-Noether theorem in historical perspective.**
*(English)*
Zbl 1060.01009

Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften 15. Berlin: Springer (ISBN 3-540-23005-X/pbk). vi, 92 p. (2005).

The present article, which is in fact a book of almost 100 pages, is based on letters and other documents of the Handschriftenabteilung of the University Library at GĂ¶ttingen concerning the Brauer-Hasse-Noether Theorem. This Theorem was published in a joint paper of R. Brauer, H. Hasse and E. Noether [J. Reine Angew. Math. 167, 399–404 (1932; Zbl 0003.24404; JFM 58.0142.03)]. It states that a central simple division algebra over a number field has a cyclic splitting field.

This Theorem is an important step in the development of the theory of simple algebras and of class field theory. It is proved with the help of the local-global principle for simple algebras stating that a simple algebra over a number field \(K\) splits if it splits locally for all places of \(K\). This principle in its cohomological formulation is so important in class field theory that nowadays some authors take it for the Brauer-Hasse-Noether Theorem. The article gives a detailed inside in the prehistory of the Theorem and the contributions to it by the three authors and by other mathematicians, in particular E. Artin and A. A. Albert. Further parts of the article consider the applications of the Theorem to topics of algebraic number theory: The structure of the Brauer group, the Grunwald-Wang Theorem and class field theory.

The study of the article is a necessity for everyone interested in the history of algebraic number theory in the time from 1930 to 1952. The reviewer finds it regrettable that the letters, which are almost all written in German, appear only in their English translation.

This Theorem is an important step in the development of the theory of simple algebras and of class field theory. It is proved with the help of the local-global principle for simple algebras stating that a simple algebra over a number field \(K\) splits if it splits locally for all places of \(K\). This principle in its cohomological formulation is so important in class field theory that nowadays some authors take it for the Brauer-Hasse-Noether Theorem. The article gives a detailed inside in the prehistory of the Theorem and the contributions to it by the three authors and by other mathematicians, in particular E. Artin and A. A. Albert. Further parts of the article consider the applications of the Theorem to topics of algebraic number theory: The structure of the Brauer group, the Grunwald-Wang Theorem and class field theory.

The study of the article is a necessity for everyone interested in the history of algebraic number theory in the time from 1930 to 1952. The reviewer finds it regrettable that the letters, which are almost all written in German, appear only in their English translation.

Reviewer: Helmut Koch (Berlin)