Why study abstract algebra? What good is it? The answer to such questions lies, of course, in the questing human intellect that admits no restraints upon its probes, nor limits to its adventures. Utility marches after discovery. The formal systems we are about to study have played important roles in the physical sciences and are of increasing importance in the biological and social sciences. While the algebra may be abstract, it nonetheless provides the setting or language for consideration of concrete physical and social problems.
Our approach to abstract algebra involves thinking of abstract as a verb form, not as an adjective. The development of algebraic structures as abstractions of the properties of integers and polynomials is shown. Thus familiar properties, observed in examples, are given a more general axiomatic setting. The intent is to involve the student in the evolution of algebraic concepts as a participant, rather than as a spectator having to assimilate lists of axiomatic descriptions. Besides the pedagogical merit of this approach, there is the mathematical merit of more closely indicating methods of mathematical research and advancement. Thus our proofs are designed to show details of arguments that later facilitate solution of exercises.

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