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.

Let A be a central simple algebra over a field F . Let k1, . . . , kr be cyclic extensions of F such that k1 ⊗F · · · ⊗F kr is a field. We investigate conditions under which A is a tensor product of symbol algebras where each ki is in a symbol F -algebra factor of the same degree as ki. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4.

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.

In this paper we associate an invariant to a biquaternion algebra B over a field K with a subfield F such that K/F is a quadratic separable extension and char(F) = 2. We show that this invariant is trivial exactly when B B 0 ⊗ K for some biquaternion algebra B 0 over F. We also study the behavior of this invariant under certain field extensions and provide several interesting examples.