Subsections

## Algebra II (MAT 535, Spring semester)

1. Linear and multilinear algebra (4 weeks)
• Minimal and characteristic polynomials. The Cayley-Hamilton Theorem.
• Similarity, Jor`dan normal form and diagonalization.
• Symmetric and antisymmetric bilinear forms, signature and diagonalization.
• Tensor products (of modules over commutative rings). Symmetric and exterior algebra (free modules). HomR(- , -) and tensor products.

References: Lang, chapters XIII and XIV; Dummit and Foote, Chapter 11.

2. Rudiments of homological algebra (2 weeks)
• Categories and functors. Products and coproducts. Universal objects, Free objects. Examples and applications.
• Exact sequences of modules. Injective and projective modules. HomR(- , -), for R a commutative ring. Extensions.

References: Lang, chapter XX; Dummit and Foote, Part V, 17.

3. Representation Theory of Finite Groups (2 weeks)
• Irreducible representations and Schur's Lemma.
• Characters. Orthogonality. Character table. Complete reducibility for finite groups. Examples.

References: Lang, chapter XVII; Dummit and Foote, Part VI; Serre.

4. Galois Theory (6 weeks)
• Irreducible polynomials and simple extensions.
• Existence and uniqueness of splitting fields. Application to construction of finite fields. The Frobenius morphism.
• Extensions: finite, algebraic, normal, Galois, transcendental.
• Galois polynomial and group. Fundamental theorem of Galois theory. Fundamental theorem of symmetric functions.
• Solvability of polynomial equations. Cyclotomic extensions. Ruler and compass constructions

### Typical References:

• S. Lang, Algebra,

• Jacobson, Basic Algebra,
2nd ed, W.H. Freeman, New York, 1985, 1989.

• Dummit and Foote, Abstract Algebra,
2nd ed, John Wiley, 1999.

• Hungerford, Algebra,
Springer-Verlag, 1974.

• B.L. van der Waerden, Algebra,
Springer-Verlag, 1994.

• Blyth, Module Theory,
Oxford University Press, 1990

• J.-P. Serre, Linear representations of finite groups,
Springer-Verlag, 1977