This book aims to cover algebraic structures and methods that play basic roles in other fields of mathematics such as algebraic geometry and representation theory. More precisely, the first chapter covers the rudiments of non-commutative rings and homological language that provide foundations for subsequent chapters. The second chapter covers commutative algebra, which we view as the local theory of algebraic geometry; the emphasis will be on (historical) connections to several complex variables. The third chapter is devoted to field theory, and the fourth to Linear algebra. The fifth chapter studies Lie algebra with emphasis on applications to arithmetic problems.

Contents

Part I. Foundations

Chapter 1. Non-commutative rings
Jacobson radical
Chapter 2. Commutative algebra
The spectrum of a ring, radical of an ideal
Chapter 3. Field theory
Galois theory, skew-field
Chapter 4. Linear algebra
Matrices over skew-fields, symplectic geometry, quadratic forms, smith normal form
Chapter 5. Lie algebras
Chapter 6. Associative algebras
Radicals of non-commutative rings, separability, central simple algebras, Hopf algebras

Part II. Applications

  • Banach algebras

Part III. Appendix

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