Cyclotomic Galois module structure and the second Chinburg invariant

1995 ◽  
Vol 117 (1) ◽  
pp. 57-82
Author(s):  
Victor P. Snaith
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Prime Power ◽  
Class Group ◽  
Number Fields ◽  
Affirmative Answer ◽  
Module Structure ◽  
Integral Group ◽  
Galois Module

AbstractWe study the second Chinburg invariant of a Galois extension of number fields. The Chinburg invariant lies in the class-group of the integral group-ring of the Galois group of the extension. A procedure is given whereby to evaluate the invariant in the case of the real cyclotomic case of regular prime power conductor and their subextensions of p-power degree. The invariant is shown to be zero in the latter cases, which yields new examples giving an affirmative answer to a question of Chinburg ([1], p. 358) which has come to be known as ‘Chinburg's Second Conjecture’ ([3], §4·2).

scholarly journals A non-abelian Stickelberger theorem

2010 ◽  
Vol 147 (1) ◽  
pp. 35-55 ◽  
Author(s):  
David Burns ◽  
Henri Johnston
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Class Group ◽  
Number Fields ◽  
Stickelberger Theorem ◽  
Use Values

AbstractLet L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring ℤ(p)[G] that annihilates the p-part of the class group of L.

scholarly journals FACTOR EQUIVALENCE OF GALOIS MODULES AND REGULATOR CONSTANTS

2014 ◽  
Vol 10 (01) ◽  
pp. 1-12
Author(s):  
ALEX BARTEL
Keyword(s):  
Number Fields ◽  
The Other ◽  
Module Structure ◽  
Integral Group ◽  
Galois Module ◽  
Group Representations ◽  
Galois Modules ◽  
Rings Of Integers ◽  
The One ◽  
Module Structures

We compare two approaches to the study of Galois module structures: on the one hand, factor equivalence, a technique that has been used by Fröhlich and others to investigate the Galois module structure of rings of integers of number fields and of their unit groups, and on the other hand, regulator constants, a set of invariants attached to integral group representations by Dokchitser and Dokchitser, and used by the author, among others, to study Galois module structures. We show that the two approaches are in fact closely related, and interpret results arising from these two approaches in terms of each other. We then use this comparison to derive a factorizability result on higher K-groups of rings of integers, which is a direct analogue of a theorem of de Smit on S-units.

Integral Representation of P-Class Groups In ℤp-Extensions and the Jacobian Variety

1998 ◽  
Vol 50 (6) ◽  
pp. 1253-1272 ◽  
Author(s):  
López-Bautista Pedro Ricardo ◽  
Gabriel Daniel Villa-Salvador
Keyword(s):  
Galois Group ◽  
Algebraic Function ◽  
Number Fields ◽  
Module Structure ◽  
Closed Field ◽  
Galois Module ◽  
Jacobian Variety ◽  
Galois Module Structure ◽  
Iwasawa Invariants ◽  
Cyclotomic Number Fields

AbstractFor an arbitrary finite Galois p-extension L/K of ℤp-cyclotomic number fields of CM-type with Galois group G = Gal(L/K) such that the Iwasawa invariants are zero, we obtain unconditionally and explicitly the Galois module structure of CL-(p), the minus part of the p-subgroup of the class group of L. For an arbitrary finite Galois p-extension L/K of algebraic function fields of one variable over an algebraically closed field k of characteristic p as its exact field of constants with Galois group G = Gal(L/K) we obtain unconditionally and explicitly the Galois module structure of the p-torsion part of the Jacobian variety JL(p) associated to L/k.

Elementary Generalizations of Hilbert's Theorem 90

1965 ◽  
Vol 8 (6) ◽  
pp. 749-757 ◽  
Author(s):  
Ian G. Connell
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Cohomology Group ◽  
Multiplicative Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
First Cohomology Group ◽  
Hilbert’S Theorem 90 ◽  
Image Position

Let K, k be fields and K|k a finite galois extension with galois group G. The multiplicative group K* of K is a G-module, that is, a module over the integral group ring ZG, the module action of an element σ ϵ G being its effect as an automorphism. It is shown in [2, p. 158] that the first cohomology group vanishes:1

scholarly journals Galois module structure of units in real biquadratic number fields

2004 ◽  
Vol 111 (2) ◽  
pp. 105-124 ◽  
Author(s):  
Marcin Mazur ◽  
Stephen V. Ullom
Keyword(s):  
Number Fields ◽  
Module Structure ◽  
Galois Module ◽  
Galois Module Structure

On Automorphisms of Complete Algebras And The Isomorphism Problem for Modular Group Rings

1990 ◽  
Vol 42 (3) ◽  
pp. 383-394 ◽  
Author(s):  
Frank Röhl
Keyword(s):  
Group Ring ◽  
Modular Group ◽  
Isomorphism Problem ◽  
Nilpotency Class ◽  
Affirmative Answer ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Analogous Question

In [5], Roggenkamp and Scott gave an affirmative answer to the isomorphism problem for integral group rings of finite p-groups G and H, i.e. to the question whether ZG ⥲ ZH implies G ⥲ H (in this case, G is said to be characterized by its integral group ring). Progress on the analogous question with Z replaced by the field Fp of p elements has been very little during the last couple of years; and the most far reaching result in this area in a certain sense - due to Passi and Sehgal, see [8] - may be compared to the integral case, where the group G is of nilpotency class 2.

ON FINITE GALOIS STABLE SUBGROUPS OF GLn IN SOME RELATIVE EXTENSIONS OF NUMBER FIELDS

2009 ◽  
Vol 08 (04) ◽  
pp. 493-503 ◽  
Author(s):  
H.-J. BARTELS ◽  
D. A. MALININ
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Galois Extension ◽  
Commutator Subgroup ◽  
Finite Extension ◽  
Number Fields ◽  
Maximal Order ◽  
Linear Groups ◽  
Stable Linear

Let K/ℚ be a finite Galois extension with maximal order [Formula: see text] and Galois group Γ. For finite Γ-stable subgroups [Formula: see text] it is known [4], that they are generated by matrices with coefficients in [Formula: see text], Kab the maximal abelian subextension of K over ℚ. This note gives a contribution to the corresponding question in the case of a relative Galois extension K/R, where R is a finite extension of the rationals ℚ. It turns out, that in this relative situation the answer to the corresponding question depends heavily on the arithmetic of the number field R, more precisely on the ramification behavior of primes in K/R. Due to the possibility of unramified extensions of R for certain number fields R there exist examples of Galois stable linear groups [Formula: see text] which are not fixed elementwise by the commutator subgroup of Gal (K/R).

scholarly journals On central commutator Galois extensions of rings

2000 ◽  
Vol 24 (5) ◽  
pp. 289-294
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Structure Theorem ◽  
Projective Group ◽  
Galois Extensions ◽  
Separable Extensions

LetBbe a ring with1,Ga finite automorphism group ofBof ordernfor some integern,BGthe set of elements inBfixed under each element inG, andΔ=VB(BG)the commutator subring ofBGinB. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and GaloisH-separable extensions. Several characterizations of a central commutator Galois extension are given. Moreover, it is shown that whenGis inner,Bis a central commutator Galois extension ofBGif and only ifBis anH-separable projective group ringBGGf. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer.

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