Instructor: Ljudmila Kamenova

Office: Math Tower 3-115
Office hours: W 4:40pm-5:40pm, TuTh 1pm-2pm in Math 3-115
e-mail: kamenova@math.sunysb.edu.

The main goal of this course is the study of fundamental concepts and methods of algebra that are used in all branches of mathematics. During the first term we cover theory of groups, rings and modules.

Additional references:

• A. Knapp, Basic Algebra, Birkhauser, 2006.
• 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.
• Rotman, Introduction to the Theory of Groups, Springer Verlag.

Midterm: Thursday, October 27th, in class.

Final: Take-home final: pick up the problems on Thursday, December 15th between 2:30 and 3:30 pm in Math Tower 3-115, return by Friday, December 16th, 3:30 pm in Math 3-115 No late finals accepted. No collaboration allowed.

HW 1 (due on Sept 8th): Write solutions to the following problems from [DF]: 1.1. Problem 22, 1.2. Problem 10, 1.3. Problems 13 and 14, 1.6. Problem 9, 1.7. Problem 12.

Look over these problems and solve them without turning them in: [DF] 0.1. Problem 7, 0.2. Problem 10, 1.1. Problem 9, 1.3. Problem 15, 1.4. Problem 11, 1.6. Problem 3

HW 2 (due on Sept 15th): Write solutions to the following problems from [DF]: 2.1. Problem 13, 2.2. Problem 7, 2.3. Problems 2 and 21, 2.4. Problem 7.

Look over these problems and solve them without turning them in: [DF] 2.1. Problems 6 and 10, 2.4. Problem 12, 2.5. Problem 11

HW 3 (due on Sept 27th): Write solutions to the following problems from [DF]: 3.1. Problem 24, 3.2. Problem 4, 3.3. Problem 8, 3.5. Problem 3.

Look over these problems and solve them without turning them in: [DF] 3.1. Problem 17, 3.2. Problem 12, 3.3. Problem 3, 3.4. Problems 7 and 8.

HW 4 (due on Oct 6th): Write solutions to the following problems from [DF]: 4.1. Problem 3, 4.2. Problem 11, 4.3. Problems 24 and 29.

HW 5 (due on Oct 13th): Write solutions to the following problems from [DF]: 4.4. Problems 2 and 7, 4.5. Problem 16.

Look over these problems and solve them without turning them in: [DF] 4.4. Problems 10 and 13, 4.5. Problems 17, 24, 26 and 37.

HW 6 (due on Oct 20th): Write solutions to the following problems from [DF]: 4.5. Problems 17, 24 and 26.

Midterm: Thursday, October 27th, in class.

HW 7 (due on Nov 3rd): Write solutions to the following problems from [DF]: 5.2. Problem 12, 5.4. Problem 15 and 5.5. Problem 8.

HW 8 (due on Nov 10th): Write solutions to the following problems from [DF]: 6.1. Problem 31, 6.3. Problem 8.

HW 9 (due on Nov 17th): Write solutions to the following problems from [DF]: 7.1. Problem 15, 7.2. Problem 4 (see Problem 3 for definition), 7.3. Problem 28.

HW 10 (due on Dec 1st): Write solutions to the following problems from [DF]: 7.4. Problem 11, 7.5. Problem 3, 7.6. Problem 5, 8.1. Problem 6, 8.2. Problem 3, 8.3. Problem 2.

HW 11 (due on Dec 8th): Write solutions to the following problems from [DF]: 9.3 Problem 3, 9.4. Problem 10, 10.1. Problem 8, 10.2 Problem 6.

1. Groups (5 weeks)
   • Direct products, Normal subgroups, Quotient groups, and the isomorphism theorems.
   • Groups acting on sets; orbits and stabilizers. Applications: class formula, centralizers and normalizers, centers of finite p-groups. Conjugacy classes of S_n
   • Sylow's Theorems, Solvable groups, Simple groups, simplicity of A_n. Examples: Finite groups of small order (<=8).
   • Structure of finitely generated abelian groups. Free groups. Applications.

   References: Lang, Chapter I; Dummit and Foote, Part I; Rotman.

2. Basic linear algebra (3 weeks)
   • Vector spaces, Linear dependence/independence, Bases, Matrices and linear maps. Dual vector space, quotient vector spaces, isomorphism theorems.
   • Determinants, basic properties. Eigenspaces and eigenvectors, characteristic polynomial.
   • Inner products and orthonormal sets. Spectral theorem for normal operators (finite dimensional case).

   References: Lang, Chapters XIII and XIV; Dummit and Foote, Chapter 11.

3. Rings, modules and algebras (6 weeks)
   • Rings, subrings, fields, ideals, homomorphisms, isomorphism theorems, polynomial rings.
   • Integral domains, Euclidean domains, PID's. UFD's and Gauss's Lemma ( F[x₁,..., x_n] is an UFD). Examples.
   • Prime ideals, maximal ideals. The Chinese remainder Theorem. Fields of fractions.
   • The Wedderburn Theorem (no proof). Simplicity and semisimplicity.
   • Noetherian rings and the Hilbert Basis Theorem.
   • Finitely generated modules over PID's, the structure theorem.

   References: Lang, Chapters II, III, V, and VI; Jacobson, Chapter 2; Dummit and Foote, Part II.

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