Algebra: Groups, Rings, and Fields

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This text presents the concepts of higher algebra in a comprehensive and modern way for self-study and as a basis for a high-level 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 never-before-published material.

From the table of contents:

• Groups: Monoids and Groups
• Cauchyís 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