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.

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).

Decomposition of Witt Rings

1982 ◽  
Vol 34 (6) ◽  
pp. 1276-1302 ◽  
Author(s):  
Andrew B. Carson ◽  
Murray A. Marshall
Keyword(s):  
Quadratic Forms ◽  
Classification Problem ◽  
Bilinear Forms ◽  
Interesting Problem ◽  
Witt Ring ◽  
Witt Rings ◽  
Finite Case ◽  
Characteristic 2 ◽  
Definition Of

We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].

Isomorphisms and Automorphisms of Witt Rings

1988 ◽  
Vol 31 (2) ◽  
pp. 250-256 ◽  
Author(s):  
David Leep ◽  
Murray Marshall
Keyword(s):  
Quadratic Forms ◽  
Multiplicative Group ◽  
Witt Ring ◽  
Witt Rings ◽  
Ring Isomorphism ◽  
Characteristic 2 ◽  
The Relationship

AbstractFor a field F, char(F) ≠ 2, let WF denote the Witt ring of quadratic forms of F and let denote the multiplicative group of 1-dimensional forms It follows from a construction of D. K. Harrison that if E, F are fields (both of characteristic ≠ 2) and ρ.WE → WF is a ring isomorphism, then there exists a ring isomorphism which “preserves dimension” in the sense that In this paper, the relationship between ρ and is clarified.

A Note on Quadratic Forms and the u-invariant

1974 ◽  
Vol 26 (5) ◽  
pp. 1242-1244 ◽  
Author(s):  
Roger Ware
Keyword(s):  
Finite Number ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Real Field ◽  
Witt Ring ◽  
Necessary And Sufficient

The u-invariant of a field F, u = u(F), is defined to be the maximum of the dimensions of anisotropic quadratic forms over F. If F is a non-formally real field with a finite number q of square classes then it is known that u ≦ q. The purpose of this note is to give some necessary and sufficient conditions for equality in terms of the structure of the Witt ring of F.

scholarly journals Abstract theory of semiorderings

2005 ◽  
Vol 72 (2) ◽  
pp. 225-250
Author(s):  
Thomas C. Craven ◽  
Tara L. Smith
Keyword(s):  
Quadratic Forms ◽  
Algebraic Theory ◽  
Covering Number ◽  
Finitely Generated ◽  
Abstract Theory ◽  
Witt Ring ◽  
Witt Rings ◽  
Covering Numbers ◽  
Spaces Of Orderings

Marshall's abstract theory of spaces of orderings is a powerful tool in the algebraic theory of quadratic forms. We develop an abstract theory for semiorderings, developing a notion of a space of semiorderings which is a prespace of orderings. It is shown how to construct all finitely generated spaces of semiorderings. The morphisms between such spaces are studied, generalising the extension of valuations for fields into this context. An important invariant for studying Witt rings is the covering number of a preordering. Covering numbers are defined for abstract preorderings and related to other invariants of the Witt ring.

scholarly journals On Witt ring of quadratIc forms.

1960 ◽  
Vol 12 (2) ◽  
pp. 187-191
Author(s):  
Yoshio NAKAMURA
Keyword(s):  
Quadratic Forms ◽  
Witt Ring

scholarly journals Topological models for stable motivic invariants of regular number rings

2022 ◽  
Vol 10 ◽  
Author(s):  
Tom Bachmann ◽  
Paul Arne Østvær
Keyword(s):  
Finite Fields ◽  
Endomorphism Ring ◽  
Quadratic Forms ◽  
Complex Numbers ◽  
Real Numbers ◽  
Witt Ring ◽  
The Real ◽  
Topological Models ◽  
Regular Number ◽  
Topological Data

Abstract For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers and finite fields. We use this to extend Morel’s identification of the endomorphism ring of the motivic sphere with the Grothendieck–Witt ring of quadratic forms to deeper base schemes.

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