## 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:**,

**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:

- 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.

**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.