(Ebook) Abstract Algebra by Paul B. Garrett ISBN 9781584886907, 1584886900

(Ebook) Abstract Algebra by Paul B. Garrett ISBN 9781584886907, 1584886900

I covered this material in a two-semester graduate course in abstract algebra in 2004-05. My goal was to rethink the standard course from scratch, ignoring traditional prejudices. The level of symbolism is as low as possible, transferring some of the load into the ambient natural language. I tried to minimize occurrence of needless common degeneracies in the natural language in which mathematics is imbedded. I tried to write proofs which would be natural outcomes of the viewpoint I promoted, with the idea that from some viewpoints there is less to remember. Robustness, as opposed to fragility, is a desirable feature of an argument. It is burdensome to remember arguments that require persistent cleverness, as opposed to a calm naturality. Of course, it is nontrivial to arrive at a viewpoint that allows proofs to seem natural. Thus, giving such proofs is revisionism. However, there are much worse revisionisms which are popular, most notably the misguided impulse to ascetic logical perfection of the development of subjects. Logical streamlining is not reliably the same as optimizing for performance. Further, logical flawlessness of an argument is not necessarily persuasive, especially if inscrutable. The worked examples are meant to be model solutions for many of the standard problems traditionally posed in such a course. I no longer believe that everyone is obliged to redo everything themselves. Hopefully it is possible for us to learn from others’ efforts. I learned abstract algebra from an early edition of S. Lang’s Algebra, from Lang’s Algebraic Number Theory, and from B.L. van der Waerden’s Algebra. These books of Lang were strongly influenced by Emil Artin and Bourbaki, and van der Waerden’s book originated in lectures of Emmy Noether.