A division algebra is a type of algebraic structure where division is possible, except by zero. It consists of a vector space equipped with a bilinear product that allows for non-zero elements to have multiplicative inverses, maintaining the essential property of associativity or alternative associativity. This structure is important in understanding how certain algebras can extend the properties of familiar number systems and is closely related to the Cayley-Dickson construction, which builds new algebras from existing ones.
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