Instructor: Ljudmila Kamenova

e-mail: kamenova@math.sunysb.edu.
Office: Math Tower 3-115
Office hours: Wed 1-4pm in Math 3-115
Grader: John Sheridan
Grader's office hours: Thu 1-2pm in Math 2-118, Tue 1-2pm and Wed 12-1pm in the MLC

Feel free to send me an e-mail or drop by my office with questions.

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 elements of homological algebra, field theory and foundations of algebraic geometry. We also study Galois theory and representations of finite groups.

Additional references:

• D. Cox, Galois Theory, Wiley-Interscience, 2004.
• 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.

HW 1 (due on Feb 1): [DF] 10.5. Problems 4, 8, 14 and 17

HW 2 (due on Feb 8): [DF] 17.1. Problems 2, 3, 4 and 5

HW 3 (due on Feb 15): [DF] 17.1. Problems 7, 10, 12 and 13

HW 4 (due on Feb 22): [DF] 13.1. Problem 8, 13.2. Problems 1, 7 and 10

HW 5 (due on March 1): [DF] 13.2. Problems 19, 20 and 21, 13.3. Problem 5

HW 6 (due on March 8): [DF] 13.4. Problem 5, 13.5. Problems 6 and 11, 13.6. Problem 8

Midterm: Monday, March 20, in class.

HW 7 (due on March 29): [DF] 14.2. Problems 3 (over Q), 7, 17 and 18

HW 8 (due on April 5): [DF] 14.3. Problems 3, 8 and 10 (p here is prime), 14.4. Problem 5

HW 9 (due on April 19): [DF] 14.6. Problems 5 (over Q), 11 and 19, 14.7. Problem 3

HW 10 (due on April 26): Click here for the problems.

Final: Thursday, May 11, in Math Tower 4-130, 11:15am-1:45pm.

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

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