Finite-Dimensional Division Algebras over Fields

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Finite-Dimensional Division Algebras over fields determine, by the Wedderburn Theorem, the semi-simple finite-dimensional algebras over a field. They lead to the definition of the Brauer group and to certain geometric objects, the Brauer-Severi varieties. The book concentrates on those algebras that have an involution. Algebras with involution appear in many contexts; they arose first in the study of the so-called multiplication algebras of Riemann matrices . The largest part of the book is the fifth chapter, dealing with involutorial simple algebras of finite dimension over a field. Of particular interest are the Jordan algebras determined by these algebras with involution;their structure is discussed. Two important concepts of these algebras with involution are the universal enveloping algebras and the reduced norm.