Varieties of Algebras
This final chapter returns to the general theory of algebras over a field. It provides a brief introduction to the theory of polynomial identities for algebras. Our main goal is to prove Amitsur’s Theorem which establishes the existence of finite dimensional central simple division algebras that are not crossed products. The choice of topics in the chapter is motivated by this objective. Fortunately, many interesting results on polynomial identities are encountered in the proof of Amitsur’s Theorem: the Amitsur—Levitzki Theorem, the existence of central polynomials, and the Kaplansky—Amitsur Theorem on primitive PI-algebras.
KeywordsDivision Algebra Polynomial Identity Quadratic Extension Surjective Homomorphism Central Simple Algebra
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