An azumaya algebra version of the kneser-tits problem for groups of type D4

L. H. Rowen; D. Saltman; Y. Segev; U. Vishne

doi:10.1080/00927870903447221

An azumaya algebra version of the kneser-tits problem for groups of type D₄

L. H. Rowen, D. Saltman, Y. Segev, U. Vishne

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We solve the following problem related to the Kneser-Tits conjecture, for Azumaya algebras. Given an Azumaya algebra D of rank 4 that is not a division algebra, whose center K is three-dimensional over the ground field F, such that cor_K/FD is trivial, is it true that every element of D having reduced norm in F is a product of n elements having both reduced norm and reduced trace in F? This is true for n ≥ 3, but false for n = 2.

Original language	English
Pages (from-to)	133-152
Number of pages	20
Journal	Communications in Algebra
Volume	39
Issue number	1
DOIs	https://doi.org/10.1080/00927870903447221
State	Published - 1 Jan 2010

Keywords

Azumaya algebra
Division algebra
Kneser-Tits problem
Quaternions
Whitehead group

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1080/00927870903447221

Cite this

@article{a6133667ccc444feae52bfb67d044ead,

title = "An azumaya algebra version of the kneser-tits problem for groups of type D4",

abstract = "We solve the following problem related to the Kneser-Tits conjecture, for Azumaya algebras. Given an Azumaya algebra D of rank 4 that is not a division algebra, whose center K is three-dimensional over the ground field F, such that corK/FD is trivial, is it true that every element of D having reduced norm in F is a product of n elements having both reduced norm and reduced trace in F? This is true for n ≥ 3, but false for n = 2.",

keywords = "Azumaya algebra, Division algebra, Kneser-Tits problem, Quaternions, Whitehead group",

author = "Rowen, {L. H.} and D. Saltman and Y. Segev and U. Vishne",

note = "Funding Information: This research was partially supported by BSF grant no. 2004-083.",

year = "2010",

month = jan,

day = "1",

doi = "10.1080/00927870903447221",

language = "English",

volume = "39",

pages = "133--152",

journal = "Communications in Algebra",

issn = "0092-7872",

publisher = "Taylor and Francis Ltd.",

number = "1",

}

TY - JOUR

T1 - An azumaya algebra version of the kneser-tits problem for groups of type D4

AU - Rowen, L. H.

AU - Saltman, D.

AU - Segev, Y.

AU - Vishne, U.

N1 - Funding Information: This research was partially supported by BSF grant no. 2004-083.

PY - 2010/1/1

Y1 - 2010/1/1

N2 - We solve the following problem related to the Kneser-Tits conjecture, for Azumaya algebras. Given an Azumaya algebra D of rank 4 that is not a division algebra, whose center K is three-dimensional over the ground field F, such that corK/FD is trivial, is it true that every element of D having reduced norm in F is a product of n elements having both reduced norm and reduced trace in F? This is true for n ≥ 3, but false for n = 2.

AB - We solve the following problem related to the Kneser-Tits conjecture, for Azumaya algebras. Given an Azumaya algebra D of rank 4 that is not a division algebra, whose center K is three-dimensional over the ground field F, such that corK/FD is trivial, is it true that every element of D having reduced norm in F is a product of n elements having both reduced norm and reduced trace in F? This is true for n ≥ 3, but false for n = 2.

KW - Azumaya algebra

KW - Division algebra

KW - Kneser-Tits problem

KW - Quaternions

KW - Whitehead group

UR - http://www.scopus.com/inward/record.url?scp=78951471697&partnerID=8YFLogxK

U2 - 10.1080/00927870903447221

DO - 10.1080/00927870903447221

M3 - Article

AN - SCOPUS:78951471697

SN - 0092-7872

VL - 39

SP - 133

EP - 152

JO - Communications in Algebra

JF - Communications in Algebra

IS - 1

ER -