2 ABSTRACT ALGEBRA on the night of May 29-30, 1832, as the radicals in Paris took to the barricades one more time, a young Frenchman, Evariste Galois, set down his final thoughts on the theory of equations and, in so doing, both laid to rest the problem posed by Lagrange and, much more important, opened the door to a new world of mathematics. In 1951, 119 years later, Hermann Weyl, one of the leaders of mathematics and physics in the early 20th century, would say in a lecture at Princeton University (February 1951): Galois' ideas, which for several decades remained a book with seven seals but later exerted a more and more profound influence upon the whole development of mathematics, are contained in a farewell letter written to a friend on the eve of his death, which he met in a silly duel at the age of twenty-one. This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind. Great ideas are the enduring legacy of mankind. The empire of Napoleon stretched briefly from Madrid to Moscow and quickly crumbled like the statue of Ozymandias. But the concepts of liberte, egalite,fraternite, the proposition that "all men are created equal," and yes, the idea of a group inspire us and shape who we are today. The principal goal of this course is to trace the history of an idea, from Lagrange to Gauss to Galois. The pursuit of this idea led mathematicians from the concrete world of polynomial equations and regular polygons to the abstract realm of groups, fields, and rings. This brave new world was so rare and strange that 100 years after Galois' death, after the theories of relativity and quantum mechanics had revolutionized physics, after another great European War and a social revolution in Russia, it would still be called Modern Algebra in the title of the influential textbook by B. L. Van der Waerden first published in 1931. Like Euclid's Elements, Van der Waerden's book represented a summation and codification of a body of knowledge that had been accumulating over the previous 160 years and had finally received an elegant and definitive treatment in the course notes of two great mathematicians of the early 20th century, Emil Artin and Emmy Noether. Like the layered cities of the ancient Near East, Van der Waerden's algebra quickly subsided into the solid foundation on which a new algebraic edifice would be built out of categories and functors, varieties and morphisms, sheaves and schemes. Thus the adjective modern became increasingly inappropriate, and it became fashionable of late for textbooks to dub this material "abstract algebra." "Abstract" is of course a relative term. Any student upon first encountering the "unknown" x in grade school will probably assure you that even this algebra (and hence all algebra) is abstract, so the adjective is redundant. At the other extreme, practitioners of late-20th-century algebra will assure you that Van der Waerden's algebra is the concrete foundation for their modern abstractions. All is relative. Following Van der Waerden, most modern "abstract algebra" texts emphasize the "abstract," beginning like Euclid with a set of axioms for a "group" or a "ring"and devel- oping in true Euclidean fashion a collection of theorems about "groups" and "rings," as well as a few examples along the way, since groups and rings are not quite so commonplace as the triangles and circles of Euclid's Elements.
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