Retrospective Outline for Math 520B, Fall Semester 1998

           Applied Galois Theory
              Cyclotomic extensions of the rational field
              Abelian and solvable extensions

           Modules
              Constructions
                 Submodule generated by a subset
                 Quotient by a submodule
                 Direct Product (categorical product)
                 Direct Sum (categorical coproduct)
                 Inductive (or direct) limit
                 Projective limit
                 Tensor Product
              Hom
              Exact sequences

           Algebras
              Constructions
                 Subalgebra generated by a subset
                 Quotient by an ideal
                 Direct Product (categorical product)
                 Relative tensor product (categorical coproduct)
                    The tensor product of polynomial rings over a field
                    The tensor product of two quotients of the base
                 Direct limit
                 Projective limit
              Freeness
              Hom
              Base extension of modules

           Modules over a PID
              Special case: the integers
              Special case: polynomials over a field
              What "is" the characteristic matrix of a given matrix?
              Structure theorem for finitely-generated modules
                 Structure theorem for finitely-generated abelian groups
                 The problem of similarity for matrices in a field
              Principal valuation rings

           When "exactness" fails
              The exact differential equation and de Rham cohomology
              Complexes and their homology
              Projective modules
              Injective modules
              Flat modules
              The concept of derived functor
              Ext and Tor

           Some topics in commutative algebra
              Valued fields and completeness
              The concept of Krull dimension
              Topological fields and local compactness
              Localization of a ring with respect to a multiplicative set
              Local rings