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 cor _K/F D 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.

Key Words:

• Azumaya algebra
• Division algebra
• Kneser–Tits problem
• Quaternions
• Whitehead group

2000 Mathematics Subject Classification:

• Primary 20G15
• Secondary 16H05

ACKNOWLEDGMENT

This research was partially supported by BSF grant no. 2004-083.

Notes

Communicated by M. Cohen.