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Advanced algebra

The course covers fundamental algebraic structures och theorems that provide a foundation for more advanced algrebraic studies, e.g., in algebraic geometry.

In particular, the following is covered.

Field theory: Splitting fields, minimal polynomials. Finite fields. Zorn's Lemma. Existence of algebraic closure and transcendental bases. Existence of maximal ideals in rings.

Module theory: Submodules and quotient modules, direct sums and products, free modules, isomorphism theorems. Finiteness conditions. Short exact sequences. Tensor products. Localization. Universal properties. Multilinear algebra. General definitions of trace and determinant. Noetherian rings and modules. The Hilbert basis theorem.

Applications: An assortment of applications within commutative algebra, representation theory, algebraic geometry and category theory.