This course has three main parts: tensor algebra,
representation theory of finite groups, and Galois theory.
Prerequisites: Basic Algebra and Linear Algebra.
Curriculum:
- Tensor product. Its universality. Canonical isomorphisms.
- Coordinates of a tensor and their transformations under change of variables.
- Tensor algebra. Symmetric and exterior algebras. Symmetric and skew-symmetric tensors.
- Simple groups. Solvable groups. Sylow’s theorems.
- Representations. Irreducible representations. Schur’s lemma.
- Semisimple algebras and modules.
- Group algebra. The Maschke theorem.
- Theory of characters.
- Algebraic extensions of fields.
- Galois extensions. Galois correspondence.
Textbooks
- D.S.Dummit, R.M.Foote, Abstract Algebra.
- Robert B.Ash, Abstract Algebra: The Basic Graduate Year.