2013 **Abstract:** Let k be a field of characteristic distinct from 2, a, a1, a2, a3 ∈ k*, D ∈ 2 Br k, exp D = 2, [Formula: see text]. We prove that D is a sum of 18 quaternion algebras. Also for a field F of certain type we construct a certain function f( ind D) such that D is a sum of f( ind D) quaternions for any D ∈ 2 Br (F).

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2014

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“…( 6) Every abelian crossed product with respect to Z n ×Z 2 is similar to the product of a symbol algebra of degree 2n and a quterinion algebra, in particular, due to Albert [1], every degree 4 algebra is similar to the product of a degree 4 symbol algebra and a quaternion algebra (Lorenz, Rowen, Reichstein, Saltman [5]). ( 7) Every abelian crossed product with respect to (Z 2 ) 4 of exponent 2 is similar to the product of 18 quaternion algebras (Sivatski [10]). ( 8) Every p-algebra of degree p n and exponent p m is similar to the product of p n − 1 cyclic algebras of degree p m (Florence [3]).…”