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21610 Algebra I Spring: 12 units The structure of finitely generated abelian groups, the Sylow theorems, nilpotent and solvable groups, simplicity of alternating and projective special linear groups, free groups, the NeilsenSchreier theorem. Vector spaces over division rings, field extensions, the fundamental Galois correspondence, algebraic closure. The Jacobson radical and the structure of semisimple rings. Time permitting, one of the following topics will be included: Wedderburn~s theorem on finite division rings, Frobenius~ Theorem. Prerequisite: Familiarity with the content of an undergraduate course on groups and rings. 