Abstract Algebra: Theory and Applications
In pre-college mathematics courses, we learned the basic methodology and notions of algebra. We appointed letters of the alphabet to abstractly represent unknown or unspecified quantities. We discovered how to translate real-world (and often complicated) problems into simple equations whose solutions, if they could be found, held the key to greater understanding. But algebra does not end there. Abstract algebra examines sets of objects (numbers, matrices, polynomials, functions, ideas) and operations on these sets. The approach is typically axiomatic: One assumes a small number of basic properties, or axioms, and attempts to deduce all other properties of the mathematical system from these. Such abstraction allows us to study, simultaneously, all structures satisfying a given set of axioms and to recognize both their commonalties and their differences. Specific topics to be covered include groups, actions, isomorphism, symmetry, permutations, rings, fields, and applications of these algebraic structures to questions outside of mathematics.