scholarly journals Realizable Galois module classes over the group ring for non abelian extensions

2013 ◽  
Vol 63 (1) ◽  
pp. 303-371 ◽  
Author(s):  
Nigel P. Byott ◽  
Bouchaïb Sodaïgui
Keyword(s):  
Group Ring ◽  
Galois Module ◽  
Abelian Extensions

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).

The p-adic Coates–Sinnott Conjecture over maximal orders

2020 ◽  
Vol 16 (06) ◽  
pp. 1227-1246
Author(s):  
Manfred Kolster ◽  
Reza Taleb
Keyword(s):  
Group Ring ◽  
Analytic Formula ◽  
Iwasawa Theory ◽  
Maximal Order ◽  
Integral Group ◽  
Abelian Extensions ◽  
Maximal Orders ◽  
Abelian Fields ◽  
Totally Real ◽  
Real Number Field

We prove the [Formula: see text]-adic version of the Coates–Sinnott Conjecture for all primes [Formula: see text], without assuming the vanishing of [Formula: see text]-invariants, for finite abelian extensions [Formula: see text] of a totally real number field [Formula: see text], where either the integral group ring [Formula: see text] of the Galois group [Formula: see text] is a maximal order in [Formula: see text] or [Formula: see text] is a CM-field of degree [Formula: see text] with [Formula: see text] odd and [Formula: see text], where the group ring [Formula: see text] is not a maximal order. The only assumption we have to make concerns the prime [Formula: see text], where for non-abelian fields we have to assume the Main Conjecture in Iwasawa theory and the equality of algebraic and analytic [Formula: see text]-invariants.

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