Real cohomology and the powers of the fundamental ideal in the Witt ring

2017 ◽  
Vol 2 (3) ◽  
pp. 357-385 ◽  
Author(s):  
Jeremy Jacobson
Keyword(s):  
Witt Ring ◽  
Fundamental Ideal

scholarly journals The Elman-Lam-Krüskemper Theorem

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Karim Johannes Becher ◽  
David B. Leep
Keyword(s):  
Elementary Proof ◽  
Stability Index ◽  
Real Field ◽  
Witt Ring ◽  
Fundamental Ideal ◽  
The Stability ◽  
Torsion Part ◽  
Torsion Free

For a (formally) real field K, the vanishing of a certain power of the fundamental ideal in the Witt ring of K(-1) implies that the same power of the fundamental ideal in the Witt ring of K is torsion free. The proof of this statement involves a fact on the structure of the torsion part of powers of the fundamental ideal in the Witt ring of K. This fact is very difficult to prove in general, but has an elementary proof under an assumption on the stability index of K. We present an exposition of these results.

scholarly journals ON HERMITIAN PFISTER FORMS

2008 ◽  
Vol 07 (05) ◽  
pp. 629-645 ◽  
Author(s):  
NICOLAS GRENIER-BOLEY ◽  
EMMANUEL LEQUEU ◽  
MOHAMMAD GHOLAMZADEH MAHMOUDI
Keyword(s):  
Quadratic Form ◽  
Division Algebra ◽  
Upper Bounds ◽  
Hermitian Forms ◽  
Witt Ring ◽  
Fundamental Ideal ◽  
Anisotropic Quadratic Form ◽  
Quaternion Division Algebra ◽  
Pure Quaternion

Let K be a field of characteristic different from 2. It is known that a quadratic Pfister form over K is hyperbolic once it is isotropic. It is also known that the dimension of an anisotropic quadratic form over K belonging to a given power of the fundamental ideal of the Witt ring of K is lower bounded. In this paper, weak analogues of these two statements are proved for hermitian forms over a multiquaternion algebra with involution. Consequences for Pfister involutions are also drawn. An invariant uα of K with respect to a nonzero pure quaternion of a quaternion division algebra over K is defined. Upper bounds for this invariant are provided. In particular an analogue is obtained of a result of Elman and Lam concerning the u-invariant of a field of level at most 2.

scholarly journals Powers of the Fundamental Ideal in the Witt Ring

2001 ◽  
Vol 239 (1) ◽  
pp. 150-160 ◽  
Author(s):  
Jón Kr. Arason ◽  
Richard Elman
Keyword(s):  
Witt Ring ◽  
Fundamental Ideal

scholarly journals Annihilating polynomials for quadratic forms

2001 ◽  
Vol 27 (7) ◽  
pp. 449-455 ◽  
Author(s):  
David W. Lewis
Keyword(s):  
Quadratic Forms ◽  
Witt Ring ◽  
Short Survey

This is a short survey of the main known results concerning annihilating polynomials for the Witt ring of quadratic forms over a field.

The Witt Ring of a Space of Ordering

1980 ◽  
Vol 258 (2) ◽  
pp. 505 ◽  
Author(s):  
Murray Marshall
Keyword(s):  
Witt Ring

Witt ring

2017 ◽  
pp. 191-206
Author(s):  
Kazimierz Szymiczek
Keyword(s):  
Witt Ring

scholarly journals The graded Witt ring and Galois cohomology. II

1989 ◽  
Vol 314 (2) ◽  
pp. 745-745 ◽  
Author(s):  
J{ón Kr. Arason ◽  
Richard Elman ◽  
Bill Jacob
Keyword(s):  
Galois Cohomology ◽  
Witt Ring

Graded Witt ring and unramified cohomology

K-Theory ◽  
1992 ◽  
Vol 6 (1) ◽  
pp. 29-44 ◽  
Author(s):  
R. Parimala ◽  
R. Sridharan
Keyword(s):  
Witt Ring ◽  
Unramified Cohomology

Quotients and Inverse Limits of Spaces of Orderings

1979 ◽  
Vol 31 (3) ◽  
pp. 604-616 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Quadratic Forms ◽  
Sums Of Squares ◽  
Inverse Limits ◽  
Witt Ring ◽  
Spaces Of Orderings ◽  
Image Position

A connection between the theory of quadratic forms defined over a given field F, and the space XF of all orderings of F is developed by A. Pfister in [12]. XF can be viewed as a set of characters acting on the group F×/ΣF×2, where ΣF×2 denotes the subgroup of F× consisting of sums of squares. Namely, each ordering P ∈ XF can be identified with the characterdefined byIt follows from Pfister's result that the Witt ring of F modulo its radical is completely determined by the pair (XF, F×/ΣF×2).

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