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This text presents the concepts of higher algebra in a comprehensive and modern way for selfstudy and as a basis for a highlevel undergraduate course. The author is one of the preeminent researchers in this field and brings the reader up to the recent frontiers of research including neverbeforepublished material. From the table of contents: Groups: Monoids and Groups Cauchys Theorem Normal Subgroups Classifying Groups Finite Abelian Groups Generators and Relations When Is a Group a Group? (Cayley's Theorem) Sylow Subgroups Solvable Groups Rings and Polynomials: An Introduction to Rings The Structure Theory of Rings The Field of Fractions Polynomials and Euclidean Domains Principal Ideal Domains Famous Results from Number Theory I Fields: Field Extensions Finite Fields The Galois Correspondence Applications of the Galois Correspondence Solving Equations by Radicals Transcendental Numbers: e and p Skew Field Theory Each chapter includes a set of exercises
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