Mon & Wedn 11:30am-12:50pm in Math Tower 4-130

**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.

- 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.

- 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.

- 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.

- 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

**Stony Brook University** expects students to maintain standards of personal integrity that
are in harmony with the educational goals of the institution; to observe national, state,
and local laws as well as University regulations; and to respect the rights, privileges,
and property of other people. Faculty must notify the Office of Judicial
Affairs of any disruptive behavior that interferes with their ability to teach,
compromises the safety of the learning environment, or inhibits students' ability to learn.

**DSS advisory.** If you have a physical, psychiatric,
medical, or learning disability that may affect your ability to carry out
the assigned course work, please contact the office of Disabled Student
Services (DSS), Humanities Building, room 133, telephone 632-6748/TDD.
DSS will review your concerns and determine what accommodations may be
necessary and appropriate. All information regarding any disability
will be treated as strictly confidential.

Students who might require special evacuation procedures in the event of an emergency are urged to discuss their needs with both the instructor and DSS. For important related information, click here.