Square-central and Artin–Schreier elements in division algebras

Abstract

We study the behavior of square-central elements and Artin-Schreier elements in division algebras of exponent 2 and degree a power of 2. We provide chain lemmas for such elements in division algebras over 2-fields F of cohomological 2-dimension cd 2 (F) ≤ 2, and deduce a common slot lemma for tensor products of quaternion algebras over such fields. We also extend to characteristic 2 a theorem proven by Merkurjev for characteristic not 2 on the decomposition of any central simple algebra of exponent 2 and degree a power of 2 over a field F with cd 2 (F) ≤ 2 as a tensor product of quaternion algebras.

On totally decomposable algebras with involution in characteristic two

“…In [1], it was also shown that if A is of degree 2 n over a field of characteristic different from 2, then A decomposes into a tensor product of quaternion algebras if and only if there exists a finite square-central subset of A (called a q-generating set) which satisfies some commuting properties. Over a field of particular cohomological dimension, it is known that central simple algebras which carry an involution of the first kind can be decomposed as a tensor product of quaternion algebras (see [12], and [4] for a characteristic 2 counterpart). In [3], a similar result was proved provided that the base field is of the u-invariant 8.…”

The descent of biquaternion algebras in characteristic two

“…In this section, we construct an example of an indecomposable algebra of exponent 2 and degree 8 over a field of 2-cohomological dimension 3. Notice that every central simple algebra of exponent 2 over F decomposes into tensor product of quaternion algebras if cd 2 (F) = 2 (see [19] if char(F) 2 and [12] if char(F) = 2). Hence, as it is the case in characteristic different from 2 [10], this example shows that 3 is the lower bound of the 2-cohomological dimension for the existence of indecomposable algebras of exponent 2.…”

Symmetric Square-Central Elements in Products of Orthogonal Involutions in Characteristic Two

“…Then z = 0 and σ(z)z = α r + α r = 0, i.e., σ| CA(x) is isotropic. 4 The set of elements represented by σ Definition 4.1. Let (A, σ) be a totally decomposable algebra with orthogonal involution over F and let α ∈ F .…”