Instructor: Ljudmila Kamenova

e-mail: kamenova@math.sunysb.edu.
Office: Math Tower 3-115
Office hours: Wednesday 4-6 p.m. in 3-115
Grader: Yi Zhu, e-mail: yzhu@math.sunysb.edu

Feel free to send me or Yi Zhu 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 linear and multilinear 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.

Midterm: Tuesday, March 15th in class

Final: Tuesday, May 17th from 2:15 p.m. to 4:45 p.m. in Physics P125

HW 1 (due on Feb 15th): [DF] 17.1. Problems 2, 3, 4 and 5

HW 2 (due on Feb 22nd): [DF] 17.1. Problems 7, 10, 12, 13 and 16

HW 3 (due on March 1st): [DF] 13.1. Problem 8, 13.2. Problems 1, 7 and 10

HW 4 (due on March 8th): [DF] 13.2. Problems 19, 20 and 22, 13.3. Problem 4

HW 5 (due on March 29th): [DF] 13.4. Problem 6, 13.5. Problems 6 and 11, 13.6. Problem 8

HW 6 (due on April 5th): [DF] 13.6. Problems 14, 15, 16 and 17, 14.1 Problem 8

HW 7 (due on April 12th): [DF] 14.2. Problems 2, 3, 5, 7 and 16

HW 8 (due on April 26th): [DF] 14.2. Problems 17, 18 and 23, 14.3. Problem 8, 14.4. Problem 5, 14.6. Problems 5 (over Q) and 19, 14.7. Problem 3

HW 9 (due on May 3rd): Click here for the problems.

HW 10 (due on May 10th): Click here for the problems.

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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