A computation of the Witt index for rational quadratic forms

1989 ◽  
Vol 38 (1) ◽  
pp. 86-98
Author(s):  
Stephen Beale ◽  
D. K. Harrison
Keyword(s):  
Quadratic Forms ◽  
Witt Index

APPLICATIONS OF SYSTEMS OF QUADRATIC FORMS TO GENERALISED QUADRATIC FORMS

2020 ◽  
Vol 102 (3) ◽  
pp. 374-386
Author(s):  
A.-H. NOKHODKAR
Keyword(s):  
Quadratic Form ◽  
Division Algebra ◽  
Quadratic Forms ◽  
Witt Index ◽  
Characteristic Two ◽  
Isometry Class

A system of quadratic forms is associated to every generalised quadratic form over a division algebra with involution of the first kind in characteristic two. It is shown that this system determines the isotropy behaviour and the isometry class of generalised quadratic forms. An application of this construction to the Witt index of generalised quadratic forms is also given.

scholarly journals The generating rank of a polar Grassmannian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ilaria Cardinali ◽  
Luca Giuzzi ◽  
Antonio Pasini
Keyword(s):  
Quadratic Forms ◽  
Vector Spaces ◽  
Hermitian Forms ◽  
Division Rings ◽  
Generating Sets ◽  
Witt Index

Abstract In this paper we compute the generating rank of k-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of k-Grassmannians arising from Hermitian forms of Witt index n defined over vector spaces of dimension N > 2n. We also study generating sets for the 2-Grassmannians arising from quadratic forms of Witt index n defined over V(N, 𝔽 q ) for q = 4, 8, 9 and 2n ≤ N ≤ 2n + 2. We prove that for N > 6 and anisotropic defect (polar corank) d ≠ 2 they can be generated over the prime subfield, thus determining their generating rank.

On the first Witt index of quadratic forms

2003 ◽  
Vol 153 (2) ◽  
pp. 455-462 ◽  
Author(s):  
Nikita A. Karpenko
Keyword(s):  
Quadratic Forms ◽  
Witt Index

scholarly journals Witt index for Galois Ring valued quadratic forms

2010 ◽  
Vol 16 (3) ◽  
pp. 175-186
Author(s):  
M.C. López-Díaz ◽  
I.F. Rúa
Keyword(s):  
Quadratic Forms ◽  
Galois Ring ◽  
Witt Index

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

Totally Wild Quadratic Forms of Type E7

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
Trace Map ◽  
Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

Linked Tori, II

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Integral Domain ◽  
Quadratic Space ◽  
Small Observation ◽  
Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

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