A Course in Algebra

The book covers all topics, which are usually included in basic courses on linear algebra and algebra in the first two years of study. In addition, it also includes a lot of non-standard and interesting material going into several different directions. The book is based on the long time teaching experience of the author. At the beginning, main algebraic structures are introduced (groups, rings, fields, algebras, vector spaces). Polynomial algebra is studied in detail and basic facts of group theory are covered. Linear algebra is covered in Chapter 2, Chapter 5 and Chapter 6, together with bilinear and quadratic functionals. Chapter 8 contains a description of general multilinear algebra. The last four chapters are devoted to commutative algebra (principal ideal domains, Noetherian rings, algebraic extensions, and affine algebraic varieties), groups (Sylow theorems, simple groups Galois extensions and Galois theory), linear representations of associative algebras (complete reducibility, invariants, division algebras) and linear Lie groups (the exponential map, the adjoint representation, basic facts on linear representations). The book is beautifully written, the choice of topics and their order is excellent and the book is very carefully produced. It contains a huge number of exercises and it appeals to geometric intuition whenever possible. It can be highly recommended for independent reading or as material for preparation of courses.

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