Power-Central Elements in Tensor Products of Symbol Algebras

Abstract

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.

On the decomposition of metabolic involutions

“…As I = I ⊥ , we have = 0. We also have dim F eA = dim F (λ+u)A = 1 2 dim F A, hence e is a metabolic idempotent. Since (λ + u)A = eA, there exist x, y ∈ A such that λ + u = ex and e = (λ + u)y. Multiplying the first equality on the left by e, we get e(λ + u) = e 2 x = ex = λ + u.…”

The descent of biquaternion algebras in characteristic two

“…Since D K is of index 4, the index of A K is also 4. So A contains a subfield isomorphic to K. A result of Merkurjev shows that there is no quaternion subalgebra of A containing K, see [11,Cor. 4.5] for char(F) 2.…”

The Arason invariant of orthogonal involutions of degree 12 and 8, and quaternionic subgroups of the Brauer group

“…Proof of Theorem 5.4. The result follows from the computation of f 3 (σ) in Proposition 2.7 and the computation of f 3 (U ) in (3). We use the same notation as in Definition 3.6, and we let h i be a rank 2 skew-hermitian form over (Q, ) such that Ad hi ≃ (Q i , ) ⊗ (H i , ) and σ = ad h1⊥h2⊥h3 .…”

“…Examples of algebras C for which condition (b) of Lemma 5.12 does not hold include indecomposable division algebras of degree 8 and exponent 2; other examples are given in [3]. Note that condition (b) is weaker than [C] ∈ Dec(M/k); it is in fact strictly weaker: see Remark 5.14.…”