Instructor: Ljudmila Kamenova
Office: Math Tower 3-115
Office hours: Tuesday 2-4 p.m. in 3-115
Grader: Jaimie Thind, office hours: Monday 4-6 p.m. in 2-122, Wednesday 10-11 in MLC, e-mail: jthind...

Feel free to send me or Jaimie Thind an e-mail or drop by outside of office hours.

The main goal of this course is to study in detail fundamental concepts and methods of algebra that are used in all branches of mathematics. During the second term we cover linear and multilinear algebra, field theory and foundations of algebraic geometry. We also study Galois theory and representations of finite groups.

Additional references:

• M. Artin, Algebra, Prentice Hall, 1991.
• S. Lang, Algebra, 3^rd ed., Addison-Wesley, 1993.
• Jacobson, Basic Algebra, 2^nd ed, W.H. Freeman, New York, 1985, 1989.
• 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, Prentice Hall, 1991.

Midterm 1: Monday, March 9th in class

Midterm 2: Friday, April 24th in class

Final: Wednesday, May 13th from 8 a.m. to 10:30 a.m. in Math 4-130

On Monday, May 4th, there will be no lecture. Check out the Clay Research Conference held at Harvard. Hironaka is going to present a resolution of singularities in positive characteristic!

HW 1 (due Feb 11): [DF] 10.1. Problem 8, 10.2. Problems 6 and 9, 10.3. Problems 9 and 11

HW 2 (due Feb 18): [DF] 10.3. Problems 24 and 27, 10.4. Problems 4, 5 and 24

HW 3 (due Feb 25): [DF] 12.1. Problems 9 and 12, 12.2. Problems 4 and 11, 12.3 Problem 12

HW 4 (due March 4): [DF] 11.3. Problems 3 and 5, 11.5. Problems 5 and 13, 10.5 Problem 1

HW 5 (due March 18): [DF] 10.5. Problems 8, 14 and 17, 17.1. Problems 3 and 4

HW 6 (due March 25): [DF] 15.1. Problems 5, 10 and 24, 15.2. Problem 8, 15.3. Problem 1

HW 7 (due April 1): [DF] 13.1. Problem 8, 13.2. Problems 1, 7, 10 and 22

HW 8 (due April 15): [DF] 13.3. Problem 4, 13.4. Problem 6, 13.5. Problems 6 and 11, 13.6. Problem 8

HW 9 (due April 29): [DF] 14.1. Problem 8, 14.2. Problems 5 and 7, 14.3. Problem 8, 14.4. Problem 5

HW 10 (due May 6): [DF] 14.5. Problems 7 and 10, 14.6. Problems 5 and 19, 14.7. Problem 3

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). Hom_R(- , -) 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. Hom_R(- , -), 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

The Instructor may be reached by e-mail at kamenova@math.sunysb.edu.

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