### MAT 535  Algebra II   Spring 2014  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'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.

Text: Abstract Algebra, by Dummit and Foote (3rd Edition), John Wiley and Sons, Inc., 2003

• D. Cox, Galois Theory, Wiley-Interscience, 2004.
• M. Artin, Algebra, Prentice Hall, 1991.
• S. Lang, Algebra, 3rd ed., Addison-Wesley, 1993.
• Jacobson, Basic Algebra, 2nd 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.

Grading: There will be two midterm tests given in class (on 3/6 and on 4/17). The final exam will be a 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. The final course grade will be determined as follows: homework = 30%, midterms = 20% each, final = 30%.

Homework:

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.

Syllabus: What follows is a tentative syllabus for the class, taken from the Graduate Handbook:
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

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.