Advanced Algebra

This course has three main parts: tensor algebra,
representation theory of finite groups, and Galois theory.

Prerequisites: Basic Algebra and Linear Algebra.

Curriculum:

1. Tensor product. Its universality. Canonical isomorphisms.
2. Coordinates of a tensor and their transformations under change of variables.
3. Tensor algebra. Symmetric and exterior algebras. Symmetric and skew-symmetric tensors.
4. Simple groups. Solvable groups. Sylow’s theorems.
5. Representations. Irreducible representations. Schur’s lemma.
6. Semisimple algebras and modules.
7. Group algebra. The Maschke theorem.
8. Theory of characters.
9. Algebraic extensions of fields.
10. Galois extensions. Galois correspondence.

Textbooks

• D.S.Dummit, R.M.Foote, Abstract Algebra.
• Robert B.Ash, Abstract Algebra: The Basic Graduate Year.