Tue & Thu 1-2:20 p.m. Math Tower 4-130

**Instructor:** Ljudmila Kamenova

**e-mail:**
kamenova@math.sunysb.edu.
**Office:** Math Tower 3-115

**Office hours:** Wed 11 am - 12 noon in the MLC, Wed 1:30 - 3:30 p.m. in
Math 3-115

**Grader:** Dingxin Zhang, e-mail: dzhang@math.sunysb.edu

**Grader's office hours:** Mon 3-4 pm in Math 2-114, Mon 4-6 pm in the MLC

Feel free to send me or Dingxin Zhang an e-mail or drop by.

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 6): 10.1. Problem 8; 10.2. Problem 6;
10.3. Problems 4 and 9.

HW **2** (due on Feb 18): 10.4. Problems 5 and 24;
10.5. Problems 8 and 17.

HW **3** (due on Feb 25): 12.1. Problems 11 and 12; 12.2. Problem 4;
12.3 Problem 12.

HW **4** (due on March 4): 17.1. Problems 2, 3, 4 and 5.

Midterm 1: Thursday, March 6, in class.

HW **5** (due on March 13): 17.1. Problems 7, 10, 12 and 13.

HW **6** (due on March 27): 15.1. Problems 13, 16, 24 and 26.

HW **7** (due on April 3): 15.2. Problems 11, 15 and 26; 15.3.
Problem 18 (in part (a) prove that I and J are radical ideals, not prime).

HW **8** (due on April 10): 13.1. Problem 8, 13.2. Problems 1, 7 and 10.

Midterm 2: Thursday, April 17, in class.

HW **9** (due on April 24): 13.2. Problems 19, 20 and 21, 13.3. Problem 5.

HW **10** (due on May 1): 13.4. Problem 5, 13.5. Problems 6 and 11,
13.6. Problem 8.

Take-home final to be picked up on 5/12 between 2 and 3 pm in Math Tower 3-115 and to be returned on 5/13 by 3pm.

**Practice problems:** 14.1. Problem 8, 14.2. Problems 2, 3, 7, 17
and 18, 14.3. Problem 8, 14.4. Problem 5, 14.6. Problem 19.

- 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

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the assigned course work, please contact the office of Disabled Student
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