Decomposable and indecomposable algebras of degree $$8$$ 8 and exponent 2

Abstract

Abstract. We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let B be a biquaternion algebra over F ( √ a) with trivial corestriction. A degree 3 cohomological invariant is defined and we show that it determines whether B has a descent to F . This invariant is used to give examples of indecomposable algebras of degree 8 and exponent 2 over a field of 2-cohomological dimension 3 and over a field M(t) where the u-invariant of M is 8 and t is an indeterminate. The construction of these indecomposable algebras uses Chow group computations provided by A. S. Merkurjev in Appendix.

The descent of biquaternion algebras in characteristic two

“…1]. Hence, Merkurjev's Chow group computations in [10,Theorem 6.7] show that the scalar extension map This theorem allows to construct an example of a biquaternion algebra with nontrivial invariant over a field of 2-cohomological dimension 3: By Theorem 4.1, every indecomposable algebra of degree 8 and exponent 2 can be scalar extended to an indecomposable algebra over a field of cohomological 2-dimension 3. Since indecomposable algebras of degree 8 and exponent 2 exist, they exist also over fields M with cd 2 (M) = 3.…”

“…In [10], the invariant δ was associated to any biquaternion algebra B over K where K/F is a quadratic field extension and char(F) 2. It was shown that the invariant is trivial exactly when B has a descent to F. This invariant proved to be a refinement of an invariant ∆ defined in [16,Section 11] for algebras of degree 8 and exponent 2, which is trivial when the algebra decomposes and is conjectured to be nontrivial otherwise.…”

“…It was then used to show the existence of at least one algebra A of degree 8 and exponent 2 with nontrivial ∆(A). The main tool Barry used in order to construct this example was [10,Proposition 4.9], whose characteristic 2 analogue is given in this paper as Proposition 6.1.…”

Invariants of degree 3 and torsion in the Chow group of a versal flag

“…By [5] the induced pullback on T -equivariant Chow groups CH T (pt) → CH T (U ) is an isomorphism. Since CH T (U ) ≃ CH(U/T ) ≃ CH(U/B) and CH T (pt) can be identified with the symmetric algebra Sym(T * ) of the group of characters of T , it gives an isomorphism (1) c CH : Sym(T * )…”

Power-Central Elements in Tensor Products of Symbol Algebras

“…As observed above, this is not true anymore in higher degree. More generally, if A is of degree 8 and exponent 2 and F ( √ d) ⊂ A is a quadratic field extension, there exists a cohomological criterion associated with the centralizer of F ( √ d) in A which determines whether F ( √ d) lies in a quaternion F -subalgebra of A (see [Ba,Prop. 4.4]).…”