Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 249A: Problem-Solving in Algebra
Summary of Sequence Content –
This sequence is intended as a workshop on solving problems in Algebra. The topics covered in the class include the following:
- Groups. Solvable groups. Semidirect products. Abelian groups. Sylow's theorems. Free groups. Group presentations.
- Rings. Chinese remainder theorem. Polynomial and group rings. Localization. Ideals. Principal ideal domains. Unique factorization domains. Hilbert's basis theorem.
- Modules. Free modules. Projective modules. Noetherian and Artinian properties. Modules over a principal ideal domain.
- Matrices. Modules over a polynomial ring. Jordan form. Bilinear forms.
- Tensor Products. Universal properties. Tensor products of modules and algebras. Symmetric and alternating algebras.
- Representations. Semisimple rings. Characters and orthogonality. Representations of finite groups (including Abelian and symmetric ones).
- Galois Theory. Algebraic, normal, separable, solvable and radical field extensions. Finite fields. Algebraic closures. Splitting fields. The fundamental theorem of Galois theory. Solvability by radicals.
- Ring extensions. Integral extensions. Noether's normalization theorem.
- Commutative Algebra. Nullstellensatz. Primary decompositions. Dedekind domains.
- Homological Algebra. Chain complexes. Homology. Exactness of functors. Long exact sequences. Injective, projective and flat resolutions.