Power-Central Elements in Tensor Products of Symbol Algebras

Let A be a central simple algebra over a field F. Let k₁,…, k_r be cyclic extensions of F such that k₁ ?_F… ?_Fk_r is a field. We investigate conditions under which A is a tensor product of symbol algebras where each field k_i lies in a symbol F-algebra factor of the same degree as k_i over F. 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.