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The Albert quadratic form for an algebra of degree four

Authors: Pasquale Mammone and Daniel B. Shapiro
Journal: Proc. Amer. Math. Soc. 105 (1989), 525-530
MSC: Primary 16A40; Secondary 11E04, 11E88, 12E15
MathSciNet review: 931742
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Abstract: Suppose $ K$ is a field and the $ K$-algebra $ A$ is expressed as a tensor product of two quaternion algebras $ A \cong {H_1} \otimes {H_2}$. Let $ {N_i}$ be the norm form on $ {H_i}$ and define the "Albert form" $ {\alpha _A}$ to be the $ 6$-dimensional quadratic form determined by $ \alpha_{A} \bot \left\langle {1, - 1} \right\rangle \cong {N_1} \bot - {N_2}$. In [Adv. in Math. 48 (1983), 149-165] Jacobson proved: (1) any two Albert forms for $ A$ are similar; (2) if $ A$ and $ B$ are algebras of this type, then $ A \cong B$ if and only if $ {\alpha _A}$ and $ {\alpha _B}$ are similar.

The authors prove this result using quadratic forms and Clifford algebras, avoiding the application of Jacobson's theory of Jordan norms.

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