Realizing dyadic factors of elementary type Witt rings and pro-2 Galois groups

1991 ◽  
Vol 208 (1) ◽  
pp. 193-208 ◽  
Author(s):  
Bill Jacob ◽  
Roger Ware
Keyword(s):  
Galois Groups ◽  
Witt Rings ◽  
Elementary Type

Graded witt rings of elementary type

1985 ◽  
Vol 272 (2) ◽  
pp. 267-280 ◽  
Author(s):  
J�n Kr. Arason ◽  
Richard Elman ◽  
Bill Jacob
Keyword(s):  
Witt Rings ◽  
Elementary Type

Abstract Witt Rings When Certain Binary Forms Represent Exactly Four Elements

1993 ◽  
Vol 45 (6) ◽  
pp. 1184-1199 ◽  
Author(s):  
Craig M. Cordes
Keyword(s):  
Group Ring ◽  
Positive Characteristic ◽  
Finitely Generated ◽  
Binary Forms ◽  
Witt Ring ◽  
Witt Rings ◽  
Elementary Type ◽  
Characteristic 2

AbstractAn abstract Witt ring (R, G) of positive characteristic is known to be a group ring S[Δ] with ﹛1﹜ ≠ Δ ⊆ G if and only if it contains a form〈1,x〉, x ≠1, which represents only the two elements 1 and x. Carson and Marshall have characterized all Witt rings of characteristic 2 which contain binary forms representing exactly four elements. Such results which show R is isomorphic to a product of smaller rings are helpful in settling the conjecture that every finitely generated Witt ring is of elementary type. Here, some special situations are considered. In particular if char(R) = 8, |D〈l, 1〉| = 4, and R contains no rigid elements, then R is isomorphic to the Witt ring of the 2-adic numbers. If char(R) = 4, |D〈l,a〉| = 4 where a ∈ D〈1, 1〉, and R contains no rigid elements, then R is either a ring of order 8 or is the specified product of two Witt rings at least one of which is a group ring. In several cases R is realized by a field.

Gorenstein Witt Rings

1988 ◽  
Vol 40 (5) ◽  
pp. 1186-1202 ◽  
Author(s):  
Robert W. Fitzgerald
Keyword(s):  
Group Ring ◽  
Quotient Ring ◽  
Injective Resolution ◽  
Gorenstein Rings ◽  
Witt Ring ◽  
Witt Rings ◽  
Local Type ◽  
Elementary Type ◽  
Ring Extensions

Throughout R is a noetherian Witt ring. The basic example is the Witt ring WF of a field F of characteristic not 2 and finite. We study the structure of (noetherian) Witt rings which are also Gorenstein rings (i.e., have a finite injective resolution). The underlying motivation is the elementary type conjecture. The Gorenstein Witt rings of elementary type are group ring extensions of Witt rings of local type. We thus wish to compare the two classes of Witt rings: Gorenstein and group ring over local type. We show the two classes enjoy many of the same properties and are, in several cases, equal. However we cannot decide if the two classes are always equal.In the first section we consider formally real Witt rings R (equivalently, dim R = 1). Here the total quotient ring of R is R-injective if and only if R is reduced. Further, R is Gorenstein if and only if R is a group ring over Z. This result appears to be somewhat deep.

scholarly journals One-Relator Maximal Pro-p Galois Groups and the Koszulity Conjectures

2020 ◽  
Author(s):  
Claudio Quadrelli
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Group Algebra ◽  
Galois Groups ◽  
Cohomology Algebra ◽  
Augmentation Ideal ◽  
Graded Algebra ◽  
Absolute Galois Group ◽  
Quadratic Algebras ◽  
Elementary Type

Abstract Let p be a prime number and let ${\mathbb{K}}$ be a field containing a root of 1 of order p. If the absolute Galois group $G_{\mathbb{K}}$ satisfies $\dim\, H^1(G_{\mathbb{K}},\mathbb{F}_p)\lt\infty$ and $\dim\, H^{\,2}(G_{\mathbb{K}},\mathbb{F}_p)=1$, we show that L. Positselski’s and T. Weigel’s Koszulity conjectures are true for ${\mathbb{K}}$. Also, under the above hypothesis, we show that the $\mathbb{F}_p$-cohomology algebra of $G_{\mathbb{K}}$ is the quadratic dual of the graded algebra ${\rm gr}_\bullet\mathbb{F}_p[G_{\mathbb{K}}]$, induced by the powers of the augmentation ideal of the group algebra $\mathbb{F}_p[G_{\mathbb{K}}]$, and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat’s elementary type conjecture.

Decomposition of Witt Rings and Galois Groups

1995 ◽  
Vol 47 (6) ◽  
pp. 1274-1289 ◽  
Author(s):  
Ján Mináč ◽  
Tara L. Smith
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Groups ◽  
Complete List ◽  
Direct Products ◽  
Free Products ◽  
Basic Part ◽  
Witt Ring ◽  
Semidirect Products ◽  
Witt Rings

AbstractTo each field F of characteristic not 2, one can associate a certain Galois group 𝒢F, the so-called W-group of F, which carries essentially the same information as the Witt ring W(F) of F. In this paper we show that direct products of Witt rings correspond to free products of these Galois groups (in the appropriate category), group ring construction of Witt rings corresponds to semidirect products of W-groups, and the basic part of W(F) is related to the center of 𝒢F. In an appendix we provide a complete list of Witt rings and corresponding w-groups for fields F with |Ḟ/Ḟ2| ≤ 16.

Witt Rings and Galois Groups

1996 ◽  
Vol 144 (1) ◽  
pp. 35 ◽  
Author(s):  
Jan Minac ◽  
Michel Spira
Keyword(s):  
Galois Groups ◽  
Witt Rings

Gorenstein Witt Rings II

1997 ◽  
Vol 49 (3) ◽  
pp. 499-519
Author(s):  
Robert W. Fitzgerald
Keyword(s):  
Classification Problem ◽  
Witt Rings ◽  
Dimension Zero ◽  
Elementary Type

AbstractThe abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classify the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type.

scholarly journals On the Cohomology of Galois Groups Determined by Witt Rings

1999 ◽  
Vol 148 (1) ◽  
pp. 105-160 ◽  
Author(s):  
Alejandro Adem ◽  
Dikran B Karagueuzian ◽  
Ján Mináč
Keyword(s):  
Galois Groups ◽  
Witt Rings
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