X ABSTRACT ALGEBRA or to give it the attention it deserves. The abstract algebra course then becomes a series of complicated finger exercises with no beautiful sonata to play at the end. Here, too, Galois Theory is the grand finale. But along the way, the students get to play many lovely preludes, nocturnes and sonatinas so even if the final sonata is never reached, the journey will have been filled with lovely music. Besides giving a more organic and evolutionary development of the subject, it is the intention of this text to emphasize the connections within algebra and between algebra and other areas of mathematics, especially geometry. The different fields of mathematics are not hermetically sealed off from each other. Quite the contrary, most of the truly important achievements in mathematics have been the product of fruitful interaction of areas. A textbook is at best a learning aid and at worst a stumbling block. Learning occurs on the dynamic interface between teacher and student. I have had the good fortune to have been inspired by many superb teachers, and wish to acknowledge a few here: Blossom Backal, who taught me high school geometry and first opened my eyes to the beauty of mathematics Ralph G. Archibald, who taught me number theory and introduced me to mathematical research. David Goldschmidt, who taught me local group theory and gave me glimpses of a truly deep thinker at work and my thesis advisor, Walter Feit, who taught me representation theory and forced me to figure out for myself why V = [y, A] 0 Cy(A) is Fitting's Lemma, With specific regard to this book, I am deeply indebted to Keith Conrad, David Pollack and Inna Korchagina for their comments, corrections and encouragement. Had I saved all of their corrections more systematically, this book would have far fewer errors than it does. Also I and the book owe much to the students I have taught (and learned from) over the years of developing this material. I mention only two representatives of the many who deserve thanks: Jon Spier and Jason Petry. Ronald Solomon
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