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Math 594: Algebra 2
Associate Professor Brian Conrad
MWF, 2:00-3:00 pm, 4096 East Hall
This course will cover the basic elements of group theory and Galois theory, as preparation for the qualifying
exam in algebra. We'll begin with a tour of the standard facts from finite group theory with an emphasis on those notions which are
important for more general groups (algebraic groups, Lie groups, etc.). This may include a brief discussion of some concepts in the
representation theory of finite groups if time permits. Once these basics are handled, we turn out attention to the theory of fields
(including characteristic p!) and the historical reason why groups were first introduced by Galois: to do Galois theory! I think that
Galois theory is one of the most awe-inspiring topics in algebra. By the end of the course, we will have completely solved several
classical problems, including how to determine which types of constructions are possible with a straightedge and compass, how to give an
`essentially' algebraic proof of the so-called Fundamental Theorem of Algebra, and how to prove that it is impossible
(in a very precise sense) to solve the general nth degree polynomial `in radicals' when n is at least 5
(and how one can derive the classical formulas for n < 5).
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