Algebras Over a Field
Algebras over a field, both associative and nonassociative, are introduced and many examples are given, among them Lie algebras and Jordan algebras. Polynomial algebras are studied in detail and results such as the division algorithm for polynomials are established. Karatsuba’s algorithm is introduced as an example of faster polynomial multiplication. Reducibility and factorization of polynomials are considered, as are the Fundamental Theorem of Algebra and its implications.