scholarly journals Classical groups over division rings of characteristic two

1972 ◽  
Vol 7 (2) ◽  
pp. 191-226 ◽  
Author(s):  
William M. Pender ◽  
G.E. Wall
Keyword(s):  
Normal Subgroup ◽  
Quadratic Form ◽  
Division Ring ◽  
Quadratic Forms ◽  
Factor Group ◽  
Classical Groups ◽  
Isometry Groups ◽  
Division Rings ◽  
Characteristic Two

The notion of quadratic form over a field of characteristic two is extended to an arbitrary division ring of characteristic two with an involution of the first kind. The resulting isometry groups are shown to have a simple normal subgroup and the structure of the factor group is calculated. It is indicated how one may define and analyse all the classical groups in a unified manner by means of quadratic forms.

scholarly journals Classical groups over division rings of characteristic two: Corrigenda and an acknowledgement

1972 ◽  
Vol 7 (2) ◽  
pp. 319-319
Author(s):  
William M. Pender
Keyword(s):  
Quadratic Forms ◽  
Clifford Algebras ◽  
Classical Groups ◽  
Division Rings ◽  
Characteristic Two

In my paper [1], part (b) of Lemma 3.4 and the remark following it are incorrect and should be omitted.The isometry P in page 215, line 2, need not be a generator of T(E, q) as asserted, but can be shown to lie in T(E, q).I am indebted to Professor Tits for pointing out that extended ideas of quadratic forms have already appeared in his paper [2] and in Wall's paper [3]. Professor Tits also discusses the corresponding groups and Clifford algebras in detail.

Subnormal Subgroups of Division Rings

1963 ◽  
Vol 15 ◽  
pp. 80-83 ◽  
Author(s):  
I. N. Herstein ◽  
W. R. Scott
Keyword(s):  
Normal Subgroup ◽  
Division Ring ◽  
Multiplicative Group ◽  
Finite Sequence ◽  
Division Rings ◽  
Subnormal Subgroups

Let K be a division ring. A subgroup H of the multiplicative group K′ of K is subnormal if there is a finite sequence (H = A0, A1, . . . , An = K′) of subgroups of K′ such that each Ai is a normal subgroup of Ai+1. It is known (2, 3) that if H is a subdivision ring of K such that H′ is subnormal in K′, then either H = K or H is in the centre Z(K) of K.

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 algebraic closure in function fields of quadratic forms in characteristic 2

1997 ◽  
Vol 55 (2) ◽  
pp. 293-297 ◽  
Author(s):  
Hamza Ahmad
Keyword(s):  
Quadratic Form ◽  
Function Field ◽  
Quadratic Forms ◽  
Function Fields ◽  
Algebraic Closure ◽  
Characteristic 2 ◽  
Characteristic Two

For a field k of characteristic not two, it is known that k is algebraically closed in the function field of any (non-degenerate) quadratic form in three or more variables. In this note we consider fields of characteristic two and decide when k is algebraically closed in a function field of a quadratic k-form. For quadratic forms in three variables this has recently been done by Ohm.

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

scholarly journals On Characteristic Morphisms: In Memoriam Thomas Macfarland Cherry

1969 ◽  
Vol 9 (3-4) ◽  
pp. 478-488 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Normal Subgroup ◽  
Factor Group ◽  
In Memoriam

This note is concerned with a translation of some concepts and results about characteristic subgroups of a group into the language of categories. As an example, consider strictly characteristic and hypercharacteristic subgroups of a group: the subgroup H of the group G is called strictly characteristic in G if it admits all ependomorphisms of G; that is all homomorphic mappings of G onto G; and H is called hypercharacteristic2 in G if it is the least normal subgroup with factor group isomorphic to G/H, that is if H is contained in every normal subgroup K of G with G/K ≅ G/H.

scholarly journals REPRESENTATIONS OF ALTERNATIVE CLIFFORD ALGEBRAS OF QUADRATIC FORMS

2014 ◽  
Vol 57 (3) ◽  
pp. 579-590 ◽  
Author(s):  
STACY MARIE MUSGRAVE
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Algebraic Structure ◽  
Quadratic Forms ◽  
Clifford Algebras ◽  
Future Research ◽  
Octonion Algebras

AbstractThis work defines a new algebraic structure, to be called an alternative Clifford algebra associated to a given quadratic form. I explored its representations, particularly concentrating on connections to the well-understood octonion algebras. I finished by suggesting directions for future research.

Isotropy of orthogonal involutions in characteristic two

2018 ◽  
Vol 17 (12) ◽  
pp. 1850240 ◽  
Author(s):  
A.-H. Nokhodkar
Keyword(s):  
Quadratic Form ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Characteristic Two ◽  
Central Simple ◽  
The Given

A totally singular quadratic form is associated to any central simple algebra with orthogonal involution in characteristic two. It is shown that the given involution is isotropic if and only if its corresponding quadratic form is isotropic.

On the existence of integral ternary quadratic forms without automorphs of finite order

2017 ◽  
Vol 26 (14) ◽  
pp. 1750102 ◽  
Author(s):  
José María Montesinos-Amilibia
Keyword(s):  
Quadratic Form ◽  
Finite Order ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Ternary Quadratic Forms

An example of an integral ternary quadratic form [Formula: see text] such that its associated orbifold [Formula: see text] is a manifold is given. Hence, the title is proved.

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