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Galois module structure. (English) Zbl 0830.11042
Providence, RI: AMS. vii, 207 p. \$ 70.00 (1994).

Over the last twenty years, the theory of Galois module structure has seen substantial progress. Originally it dealt with the study of $G\left(L/K\right)$-modules ${𝒪}_{L}$ for a Galois extension $L/K$ of global number fields. Fröhlich’s conjecture states that the class of ${𝒪}_{L}$ in the class group $Cl\left({ℤ}_{K}\left[G\left(L/K\right)\right]\right)={K}_{0}\left(ℤ\left[G\left(L/K\right)\right]\right)/\left\{\text{free}\phantom{\rule{4.pt}{0ex}}\text{modules}\right\}$ for a tamely ramified extension $L/K$ coincides with the class of an analytic invariant ${W}_{L/K}$ related with the Artin root number. It was M. J. Taylor [Invent. Math. 63, 41-79 (1981; Zbl 0469.12003)] who proved this conjecture in 1981. Later T. Chinburg stated several generalizations of this conjecture using Chinburg invariants ${\Omega }\left(L/K,i\right)$. For example, Fröhlich-Chinburg’s conjecture states that for a Galois extension $L/K$ the second Chinburg invariant ${\Omega }\left(L/K,2\right)\in Cl\left(ℤ\left[G\left(L/K\right)\right]\right)$ coincides with the class of ${W}_{L/K}$.

Very different from the book of A. Fröhlich [Galois module structure of algebraic integers, Springer-Verlag (1983; Zbl 0501.12012)], this book is devoted to discussions of these conjectures and corresponding techniques. It is based on a graduate course given by the author at the Fields Institute in 1993. There are seven chapters and several dozen exercises.

The first chapter contains many preliminaries in a brief form. In particular, a method of explicit Brauer induction, developed by the author and R. Boltje, is described there. It seems to be very useful in this subject in numerous applications throughout the book.

The second chapter deals with the class group of an integral group-ring and the second Chinburg invariant. The Fröhlich-Chinburg conjecture is verified for some subextensions of cyclotomic extensions using $p$-adic $L$-functions. The third chapter introduces logarithmic technique which was originally discovered by M. J. Taylor and R. Oliver. Here the explicit Brauer induction provides a simple construction of the group- ring logarithm. Based on a new approach of congruence technology the fourth chapter exposes a sketch of the M. J. Taylor proof of the Fröhlich conjecture in the special case of a $p$-group of odd order.

Chapter V deals with an almost complete proof of D. Holland’s result that the Fröhlich-Chinburg conjecture holds in the class-group of the maximal order. For the proof a method of canonical factorization due to Holland is described. The sixth chapter is devoted to quaternion considerations of the Fröhlich-Chinburg conjecture, and the seventh chapter contains a construction of new class-group invariants of the Chinburg type arising from cohomological studies of the higher algebraic $K$-groups of rings of $S$-integers.

The book offers a fairly fast presentation of important results in Galois module structure. The subject is growing, and many new results and conjectures appeared after its publication. Being the first book on this thriving subject of such scope, it will be very useful for specialists inside and outside of this subject, as well as for graduate students.

##### MSC:
 11R33 Integral representations related to algebraic numbers 11-02 Research monographs (number theory) 11R70 $K$-theory of global fields 11S25 Galois cohomology for local fields 11R29 Class numbers, class groups, discriminants 11S31 Class field theory for local fields; $p$-adic formal groups 11R34 Galois cohomology for global fields 19F27 Étale cohomology, higher regulators, zeta and $L$-functions ($K$-theory) 11S40 Zeta functions and $L$-functions of local number fields