MODERN ALGEBRA based on the author’s lectures on a one-semester course of abstract algebra at Fudan University, Shanghai, China is a concise introduction to algebraic structures up to basic Galois theory. The first four chapters cover the basic materials on groups, rings, fields, polynomials and linear algebra over arbitrary fields. The remaining four chapters include more advanced topics such as Sylow theorem, structure of finitely generated abelian groups, solvable groups, field extensions, finite fields and elementary Galois Theory. Ruler and compass construction and insolvability of quintic equations are included as applications of algebra. In the section on ring theory, the emphasis is on commutative rings and fields. The text is written for undergraduate students in math major who have taken a course of linear algebra. The brief solutions or hints of more than 150 exercises are provided for the convenience of the readers.

Foreword / Preliminaries and Notations / Elements of Groups: Definitions and Examples / Subgroups / Permutation Groups / Cosets / Normal Subgroups and Quotient Groups / Alternating Groups / Homomorphisms of Groups / Direct Product of Groups / Automorphisms of Finite Cyclic Groups and the Euler Function / Group Action / Elements of Rings and Fields: Basic Definitions / Ideals and Quotient Rings / Homomorphisms of Rings / Elementary Properties of Fields / Polynomials and Rational Functions: Polynomials in One Variable / Division Algorithm / Polynomials in Several Variables / Factorization / Polynomial Functions / Vector Spaces: Vector Spaces and Linear Transformations / Quotient Spaces / Topics in Group Theory: The Orbit Formula of an Action by a Finite Group / Sylow Subgroups / Structure of Finitely Generated Abelian Groups / Solvable Groups / Field Extensions: Definitions and First Properties of Field Extensions / Algebraic Extensions / Constructions of Field Extensions / Algebraically Closed Field / Ruler and Compass Construction / Finite Fields: Basic Theory / The Structure of Multiplicative Group of a Finite Field / Finite Galois Theory / Basic Theory / Solvable Extension and Solvability of Algebraic Equations by Radicals / Appendices / Bibliography / Index.

Under and Postgraduate Students, Researchers and Teachers