# 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

Approved: 2010-10-01,
Suggested Textbook: (actual textbook varies by instructor; check your instructor)

Prerequisites:
Courses MAT 250AB or consent of instructor.
Course Description:
Sequence Description - Problem-solving in graduate algebra: groups, rings, modules, matrices, tensor products, representations, Galois theory, ring extensions, commutative algebra and homological algebra.
Suggested Schedule:

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:

1. Groups. Solvable groups. Semidirect products. Abelian groups. Sylow's theorems. Free groups. Group presentations.
2. Rings. Chinese remainder theorem. Polynomial and group rings. Localization. Ideals. Principal ideal domains. Unique factorization domains. Hilbert's basis theorem.
3. Modules. Free modules. Projective modules. Noetherian and Artinian properties. Modules over a principal ideal domain.
4. Matrices. Modules over a polynomial ring. Jordan form. Bilinear forms.
5. Tensor Products. Universal properties. Tensor products of modules and algebras. Symmetric and alternating algebras.
6. Representations. Semisimple rings. Characters and orthogonality. Representations of finite groups (including Abelian and symmetric ones).
7. 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.
8. Ring extensions. Integral extensions. Noether's normalization theorem.
9. Commutative Algebra. Nullstellensatz. Primary decompositions. Dedekind domains.
10. Homological Algebra. Chain complexes. Homology. Exactness of functors. Long exact sequences. Injective, projective and flat resolutions.
Assessment:
249A (Spring) and 249B (the following Fall) are offered as a sequence with deferred grading in Spring. The final grade for the sequence will be recorded at the end of Fall. 249A will have 1 unit; 249B will have 2 units. May be repeated once.