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: Xin Zhang, e-mail: xzhang@math.sunysb.edu

Feel free to send me or Xin 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 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: Thursday, March 22nd in class

Final: Take-home final. Pick up the problems on Monday, May 7th between 2:30 and 3:30 p.m. in my office (Math 3-115). Return your final on Tuesday, May 8th by 3:30 p.m. in Math 3-115.

HW 1 (due on February 2nd): [DF] 10.5. Problems 8, 14 and 17

HW 2 (due on February 9th): [DF] 17.1. Problems 2, 3, 4 and 5

HW 3 (due on February 16th): [DF] 17.1. Problems 7, 10, 12 and 13

HW 4 (due on February 23rd): [DF] 13.1. Problem 8, 13.2. Problems 1, 7 and 10

HW 5 (due on March 1st): [DF] 13.2. Problems 19, 20 and 21, 13.3. Problems 4 and 5

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

HW 7 (due on March 15th) [DF] 13.6. Problems 14, 15, 16 and 17, 14.1. Problem 8

Midterm: Thursday, March 22nd, in class.

HW 8 (due on April 10th) [DF] 14.2. Problems 2, 3, 7, 17 and 18

HW 9 (due on April 12th) [DF] 14.3. Problem 8, 14.4. Problem 5, 14.6. Problems 5 (over Q) and 19, 14.7. Problem 3

HW 10 (due on April 19th) [DF] 12.1. Problems 9, 11 and 12

HW 11 (due on April 26th) [DF] 12.2. Problem 4, 12.3. Problem 12

HW 12 (due on May 3rd): 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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