scholarly journals Four-fold Massey products in Galois cohomology

2018 ◽  
Vol 154 (9) ◽  
pp. 1921-1959 ◽  
Author(s):  
Pierre Guillot ◽  
Ján Mináč ◽  
Adam Topaz
Keyword(s):  
Number Field ◽  
Galois Cohomology ◽  
Number Fields ◽  
Galois Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Massey Products ◽  
Necessary And Sufficient ◽  
Global Principle

In this paper, we develop a new necessary and sufficient condition for the vanishing of $4$-Massey products of elements in the modulo-$2$ Galois cohomology of a field. This new description allows us to define a splitting variety for $4$-Massey products, which is shown in the appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such $4$-Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.

scholarly journals Note on linear relations in Galois cohomology and étale K-theory of curves

2021 ◽  
pp. 2150010
Author(s):  
Piotr Krasoń
Keyword(s):  
Abelian Variety ◽  
Galois Cohomology ◽  
Number Fields ◽  
Sufficient Condition ◽  
Linear Relations ◽  
Tate Module ◽  
K Theory ◽  
Weil Type ◽  
Global Principle

In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [G. Banaszak and P. Krasoń, On a local to global principle in étale K-groups of curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for étale [Formula: see text]-theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases, this result is the best possible i.e. if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for étale [Formula: see text]-theory of a curve. The dynamical local to global principle for the groups of Mordell–Weil type has recently been considered by S. Barańczuk in [S. Barańczuk, On a dynamical local-global principle in Mordell-Weil type groups, Expo. Math. 35(2) (2017) 206–211]. We show that all our results remain valid for Quillen [Formula: see text]-theory of [Formula: see text] if the Bass and Quillen–Lichtenbaum conjectures hold true for [Formula: see text]

scholarly journals On the module structure of the ring of all integers of a p-adic number field

1974 ◽  
Vol 54 ◽  
pp. 53-59 ◽  
Author(s):  
Yoshimasa Miyata
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Module Structure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Prime Degree ◽  
Necessary And Sufficient

Let k be a p-adic number field and o be the ring of all integers of k. Let K/k be a cyclic ramified extension of prime degree p with Galois group G. Then the ring of all integers of K is o[G]-module. The purpose of this paper is to give a necessary and sufficient condition for o[G]-modale to be indecomposable.

scholarly journals One-Valued Mappings of Groups into Fields

1953 ◽  
Vol 6 ◽  
pp. 41-52 ◽  
Author(s):  
Katsuhiko Masuda
Keyword(s):  
Finite Groups ◽  
Galois Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Galois Fields ◽  
Necessary And Sufficient

The aim of this article is to investigate algebraic nature of systems of one-valued mappings of given group into given field and to apply it to the theory of Galois algebras and duality of compact T0-groups. The results obtained in the following are those; factor systems of Galois algebras with finite Galois groups are defined without any restrictions on the orders of Galois groups and the coefficient fields, a necessary and sufficient condition for them to be associated with Galois fields is obtained, dualities of finite groups are obtained very simply without any restrictions for coefficient field of representations, and Tannaka’s duality of compact T0-groups is proved without the use of the compactness of Tannaka representation groups of representations of compact T- groups and the use of Kampen’s theorem.

scholarly journals On linear independence of trigonometric numbers

2018 ◽  
Vol 34 (2) ◽  
pp. 157-166
Author(s):  
ARNO BERGER ◽  
Keyword(s):  
Linear Independence ◽  
Number Fields ◽  
Rational Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

A necessary and sufficient condition is established for 1, cos(πr1), and cos(πr2) to be rationally independent, where r1, r2 are rational numbers. The elementary computational argument yields linear independence over larger number fields as well.

scholarly journals On the module structure in a cyclic extension over a -adic number field

1979 ◽  
Vol 73 ◽  
pp. 61-68 ◽  
Author(s):  
Yoshimasa Miyata
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Cyclic Extension ◽  
Module Structure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Let p be a prime. Let k be a -adic number field and be the ring of all integers of k. Let K/k be a cyclic totally ramified extension of degree pn with Galois group G. Clealy the ring of all integers of K is an [G]-module, and the purpose of this paper is to give a necessary and sufficient condition for the [G]-module to be indecomposable.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

Sign in / Sign up

Export Citation Format

Share Document