The Galois Module Structure of the Integers in Wildly Ramified Extensions PDF Download
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Author: Gove Griffith Elder
Publisher:
ISBN:
Category :
Languages : en
Pages : 224
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Author: Gove Griffith Elder
Publisher:
ISBN:
Category :
Languages : en
Pages : 224
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Book Description
Author: Lindsay Childs
Publisher: American Mathematical Soc.
ISBN: 0821821318
Category : Mathematics
Languages : en
Pages : 225
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Book Description
This book studies Hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. The results, not previously published in book form, show that Hopf algebras play a natural role in local Galois module theory. Included in this work are expositions of short exact sequences of Hopf algebras; Hopf Galois structures on separable field extensions; a generalization of Noether's theorem on the Galois module structure of tamely ramified extensions of local fields to wild extensions acted on by Hopf algebras; connections between tameness and being Galois for algebras acted on by a Hopf algebra; constructions by Larson and Greither of Hopf orders over valuation rings; ramification criteria of Byott and Greither for the associated order of the valuation ring of an extension of local fields to be Hopf order; the Galois module structure of wildly ramified cyclic extensions of local fields of degree p and p2; and Kummer theory of formal groups. Beyond a general background in graduate-level algebra, some chapters assume an acquaintance with some algebraic number theory. From there, this exposition serves as an excellent resource and motivation for further work in the field.
Author: Victor Percy Snaith
Publisher: American Mathematical Soc.
ISBN: 9780821871782
Category : Mathematics
Languages : en
Pages : 220
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Book Description
This is the first published graduate course on the Chinburg conjectures, and this book provides the necessary background in algebraic and analytic number theory, cohomology, representation theory, and Hom-descriptions. The computation of Hom-descriptions is facilitated by Snaith's Explicit Brauer Induction technique in representation theory. In this way, illustrative special cases of the main results and new examples of the conjectures are proved and amplified by numerous exercises and research problems.
Author: Victor Percy Snaith
Publisher: American Mathematical Soc.
ISBN: 082180264X
Category : Mathematics
Languages : en
Pages : 218
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Book Description
Galois module structure deals with the construction of algebraic invariants from a Galois extension of number fields with group $G$. This title addresses the Chinburg conjectures. It provides the background in algebraic and analytic number theory, cohomology, representation theory, and Hom-descriptions.
Author: Henri Louis Alistair Johnston
Publisher:
ISBN:
Category :
Languages : en
Pages : 124
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Book Description
Author: Martin John Taylor
Publisher:
ISBN:
Category : Galois theory
Languages : en
Pages : 162
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Book Description
Author: A. Fröhlich
Publisher: Springer Science & Business Media
ISBN: 3642688160
Category : Mathematics
Languages : en
Pages : 271
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Book Description
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.
Author: A. Fröhlich
Publisher: Springer
ISBN: 9783642688188
Category : Mathematics
Languages : en
Pages : 266
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Book Description
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.
Author: Sugil Lee
Publisher:
ISBN:
Category : Electronic dissertations
Languages : en
Pages : 59
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Book Description
The main theme of our study is the obstruction to the existence of a normal integral basis for certain Galois modules of geometric origin. When G is a finite group acting on a projective scheme X over \\Spec Z and F is a G-equivariant coherent sheaf of O_X-modules, the sheaf cohomology groups H. i(X, \\F) are G-modules, and one asks if its equivariant Euler characteristic$$\\chi(X, F) := \\sum_i (-1). i [H. i(X, F)]$$can be calculated using a bounded complex of finitely generated free modules over Z[G]. Then we say that the cohomology of F has a normal integral basis. The obstruction to the existence of a normal integral basis has been of great interest in the classical case of number fields: As conjectured by Frohlich and proven by Taylor, when N/Q is a finite tamely ramified Galois extension with Galois group G, the Galois module structure of the ring of integers O_N is determined (up to stable isomorphism) by the root numbers appearing in the functional equations of Artin L-functions associated to symplectic representations of G. Chinburg started a generalization of the theory to some schemes with tame group actions by introducing the reduced projective Euler characteristic classes $\\overline{\\chi}. P(X, F)$.These Euler characteristics are elements of the class group $Cl(Z[G])$ and give the obstruction to the existence of normal integral basis.Our aim is to generalize the theory to the ``simplest'' kind of wild ramification, namely to weakly ramified covers of curves over Spec Z. If N/Q is wildly ramified, then O_N is not a free Z[G]-module. Erez showed that when the order |G| is odd, then the different ideal $\\frak{D}_{N/Q}$ is a square, and the square root of the inverse different is a locally free Z[G]-module if and only if N/Q is weakly ramified. Kock classified all fractional ideals of weakly ramified local rings that have normal integral bases. We generalize both of the results to curves over Spec Z to construct projective Euler characteristic for certain equivariant sheaves on weakly ramified covers of curves.
Author: Alfred Weiss
Publisher: American Mathematical Soc.
ISBN: 0821802658
Category : Mathematics
Languages : en
Pages : 106
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Book Description
This text is the result of a short course on the Galois structure of S -units that was given at The Fields Institute in the autumn of 1993. Offering a new angle on an old problem, the main theme is that this structure should be determined by class field theory, in its cohomological form, and by the behaviour of Artin L -functions at s = 0. A proof of this - or even a precise formulation - is still far away, but the available evidence all points in this direction. The work brings together the current evidence that the Galois structure of S -units can be described. This is intended for graduate students and research mathematicians, specifically algebraic number theorists.
