##
**Field theory: classical foundations and multiplicative groups.**
*(English)*
Zbl 0677.12010

The author declares a two-fold purpose in writing this book. The first is to provide a young algebraist with a reasonably comprehensive summary of that material on which research in field theory is based, and the second is to indicate how this material is applied to attack problems of current research into the structure and classification of multiplicative groups of fields. Thus the first three chapters presuppose only the most elementary ideas of group and ring theory and go on to present the sort of topics that might be expected in a text on field theory together with a few gems which include material that has appeared during the last four or so years. The final chapter of about 100 pages is rather more specialized in its study of the multiplicative groups of fields and contains some material not previously published.

The first chapter starts by assuming only such elementary notions of group and ring theory as factor groups and rings, direct products and p- groups. It goes on to define and study briefly PID’s, GCD’s, UFD’s, polynomial rings, tensor products of modules and algebras and ends with topological groups and rings. Here the presentation is brief and once or twice clarity is sacrificed to brevity.

Chapter 2 occupies more than half of the text. Here the aim is to introduce and develop classical topics with some recent additional refinements to the well established theory. The first few sections run through the usual notions of elementary Galois theory allowing infinite extensions when possible and including Artin’s proof of the existence of an algebraic closure. For the ring of integers R of quadratic extensions \({\mathbb{Q}}(\sqrt{d})\) and pure cubic extensions \({\mathbb{Q}}(^ 3\sqrt{d})\), the structure of the group of units U(R) is studied in some detail; and in both cases, for certain d, algorithms are included for the construction of the canonical fundamental unit. The main theorem of infinite Galois theory is proved giving the one-one correspondence between intermediate fields K of a Galois extension E/F and subgroups of \(Gal(E/F)\) closed in the Krull topology. Consideration is given to special extensions including cyclic extensions, Kummer extensions, radical extensions and abelian p-extensions (of a field of characteristic p). For this last one a digression for the construction of the Witt ring is required. Some recent results included are:

(i) Kneser’s theorem giving conditions for \(| F^*M/F^*| =(F(M):F)\) when M is a subgroup of the multiplicative group \(E^*\) of E/F;

(ii) Schinzel’s theorem which gives conditions for the Galois group of \(X^ n-a\) to be abelian; and

(iii) a theorem of Isaacs which gives sufficient conditions for \((F(\alpha +\beta):F)=(F(\alpha):F)(F(\beta):F).\)

A good inroduction to Galois cohomology is included, enough to interpret \(H^ 2(G;E^*)\) and \(H^ 3_ 0(G;E^*)\) in terms of Brauer groups. The chapter ends by introducing the cogalois theory of radical extensions due to Greither and Harrison, and the theory of formally real fields of Artin and Schreier.

Valuation theory is dealt with in chapter 3. Here the author has restricted himself to those topics which will be of assistance in the final chapter. He includes Dedekind (but not Krull) domains, the notion of valuation ring and place, the extension of a valuation and the idea of ramification with applications to algebraic number fields. The chapter ends with the characterization of local fields and an explicit formula for the inertia field of a finite extension of the rational p-adic field.

This reader was left with a feeling that the real reason for writing the text is to draw together those results in research notes concerning the group structure of the multiplicative group \(F^*\) of a field F. This is done in the final chapter. The preliminaries include the characterization of \(F^*\) when F is the quotient field of a UFD or a Dedekind domain, and less trivially a proof of the Dirichlet-Chevalley-Hasse unit theorem. The structure of the torsion group \(t(F^*)\) is given for general F, and then of \(F^*\) itself when F is global, algebraically closed, real closed, rational p-adic or local. For fields \(E\supset F\), \(E^*/F^*\) is identified when E, F are algebraic number fields and is shown not to be finitely generated when F is infinite (Brandis’ theorem). The last four sections deal with the extent to which the property \(E^*/t(E^*)\) is free for finite extensions E/F is inherited by other extensions, this work is mainly due to May and includes results previously unpublished.

The first chapter starts by assuming only such elementary notions of group and ring theory as factor groups and rings, direct products and p- groups. It goes on to define and study briefly PID’s, GCD’s, UFD’s, polynomial rings, tensor products of modules and algebras and ends with topological groups and rings. Here the presentation is brief and once or twice clarity is sacrificed to brevity.

Chapter 2 occupies more than half of the text. Here the aim is to introduce and develop classical topics with some recent additional refinements to the well established theory. The first few sections run through the usual notions of elementary Galois theory allowing infinite extensions when possible and including Artin’s proof of the existence of an algebraic closure. For the ring of integers R of quadratic extensions \({\mathbb{Q}}(\sqrt{d})\) and pure cubic extensions \({\mathbb{Q}}(^ 3\sqrt{d})\), the structure of the group of units U(R) is studied in some detail; and in both cases, for certain d, algorithms are included for the construction of the canonical fundamental unit. The main theorem of infinite Galois theory is proved giving the one-one correspondence between intermediate fields K of a Galois extension E/F and subgroups of \(Gal(E/F)\) closed in the Krull topology. Consideration is given to special extensions including cyclic extensions, Kummer extensions, radical extensions and abelian p-extensions (of a field of characteristic p). For this last one a digression for the construction of the Witt ring is required. Some recent results included are:

(i) Kneser’s theorem giving conditions for \(| F^*M/F^*| =(F(M):F)\) when M is a subgroup of the multiplicative group \(E^*\) of E/F;

(ii) Schinzel’s theorem which gives conditions for the Galois group of \(X^ n-a\) to be abelian; and

(iii) a theorem of Isaacs which gives sufficient conditions for \((F(\alpha +\beta):F)=(F(\alpha):F)(F(\beta):F).\)

A good inroduction to Galois cohomology is included, enough to interpret \(H^ 2(G;E^*)\) and \(H^ 3_ 0(G;E^*)\) in terms of Brauer groups. The chapter ends by introducing the cogalois theory of radical extensions due to Greither and Harrison, and the theory of formally real fields of Artin and Schreier.

Valuation theory is dealt with in chapter 3. Here the author has restricted himself to those topics which will be of assistance in the final chapter. He includes Dedekind (but not Krull) domains, the notion of valuation ring and place, the extension of a valuation and the idea of ramification with applications to algebraic number fields. The chapter ends with the characterization of local fields and an explicit formula for the inertia field of a finite extension of the rational p-adic field.

This reader was left with a feeling that the real reason for writing the text is to draw together those results in research notes concerning the group structure of the multiplicative group \(F^*\) of a field F. This is done in the final chapter. The preliminaries include the characterization of \(F^*\) when F is the quotient field of a UFD or a Dedekind domain, and less trivially a proof of the Dirichlet-Chevalley-Hasse unit theorem. The structure of the torsion group \(t(F^*)\) is given for general F, and then of \(F^*\) itself when F is global, algebraically closed, real closed, rational p-adic or local. For fields \(E\supset F\), \(E^*/F^*\) is identified when E, F are algebraic number fields and is shown not to be finitely generated when F is infinite (Brandis’ theorem). The last four sections deal with the extent to which the property \(E^*/t(E^*)\) is free for finite extensions E/F is inherited by other extensions, this work is mainly due to May and includes results previously unpublished.

Reviewer: D.Kirby

### MSC:

12E99 | General field theory |

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |

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