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In this volume we present a survey of the theory of Galois module
structure for rings of algebraic integers. This theory has
experienced a rapid growth in the last ten to twelve years,
acquiring mathematical depth and significance and leading to new
insights also in other branches of algebraic number theory. The
decisive take-off point was the discovery of its connection with
Artin L-functions. We shall concentrate on the topic which has been
at the centre of this development, namely the global module
structure for tame Galois extensions of numberfields -in other
words of extensions with trivial local module structure. The basic
problem can be stated in down to earth terms: the nature of the
obstruction to the existence of a free basis over the integral
group ring ("normal integral basis"). Here a definitive pattern of
a theory has emerged, central problems have been solved, and a
stage has clearly been reached when a systematic account has become
both possible and desirable. Of course, the solution of one set of
problems has led to new questions and it will be our aim also to
discuss some of these. We hope to help the reader early on to an
understanding of the basic structure of our theory and of its
central theme, and to motivate at each successive stage the
introduction of new concepts and new tools.
This book originates from graduate courses given in Cambridge and
London. It provides a brisk, thorough treatment of the foundations
of algebraic number theory, and builds on that to introduce more
advanced ideas. Throughout, the authors emphasise the systematic
development of techniques for the explicit calculation of the basic
invariants, such as rings of integers, class groups, and units.
Moreover they combine, at each stage of development, theory with
explicit computations and applications, and provide motivation in
terms of classical number-theoretic problems. A number of special
topics are included that can be treated at this level but can
usually only be found in research monographs or original papers,
for instance: module theory of Dedekind domains; tame and wild
ramifications; Gauss series and Gauss periods; binary quadratic
forms; and Brauer relations. This is the only textbook at this
level which combines clean, modern algebraic techniques together
with a substantial arithmetic content. It will be indispensable for
all practising and would-be algebraic number theorists.
Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer
Book Archives mit Publikationen, die seit den Anfangen des Verlags
von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv
Quellen fur die historische wie auch die disziplingeschichtliche
Forschung zur Verfugung, die jeweils im historischen Kontext
betrachtet werden mussen. Dieser Titel erschien in der Zeit vor
1945 und wird daher in seiner zeittypischen politisch-ideologischen
Ausrichtung vom Verlag nicht beworben.
Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer
Book Archives mit Publikationen, die seit den Anfangen des Verlags
von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv
Quellen fur die historische wie auch die disziplingeschichtliche
Forschung zur Verfugung, die jeweils im historischen Kontext
betrachtet werden mussen. Dieser Titel erschien in der Zeit vor
1945 und wird daher in seiner zeittypischen politisch-ideologischen
Ausrichtung vom Verlag nicht beworben.
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