Instructor: Ljudmila Kamenova
Office: Math Tower 3-115
Office hours: TTh 2:30p.m. - 3:30p.m. in Math Tower 3-115, W 1:30p.m. - 2:30p.m. in the Math Learning Center

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 first term we cover group, ring and module theories.

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.

HW 1 (due Sept 11): [DF] 0.1. Problem 7, 0.2. Problem 10, 0.3. Problem 4

HW 2 (due Sept 18): [DF] 1.1. Problem 9, 1.2. Problem 18, 1.3. Problem 14, 1.4. Problem 7

HW 3 (due Sept 25): [DF] 1.3. Problem 15, 1.6. Problem 3, 1.7. Problem 12, 2.1. Problem 13, 2.2. Problem 7

HW 4 (due Oct 7): [DF] 2.3. Problem 2, 3.1. Problem 24, 3.2. Problem 4, 3.5. Problem 3, 4.1. Problem 3

HW 5 (due Oct 16): [DF] 4.2. Problem 11, 4.3. Problem 29, 4.4. Problem 2, 5.1. Problem 18

HW 6 (due Oct 23): [DF] 5.2. Problems 6 and 12, 4.5. Problems 16, 17 and 26, 5.5. Problem 8

HW 7 (due Nov 6): [DF] 6.1. Problem 31, 11.1. Problems 4 and 13, 11.2 Problems 11 and 37

HW 8 (due Nov 20): [DF] 7.1. Problem 15, 7.2. Problem 4, 7.3. Problem 28 (see problem 26 for the definition of characteristic), 7.4. Problems 10 and 11

HW 9 (due Dec 4): [DF] 7.5. Problems 2, 3 and 4, 7.6. Problems 8 and 9

HW 10 (due Dec 11): [DF] 8.1. Problem 8, 8.2. Problem 5.

Final: Take-home final. To be given on Wednesday, December 17th beween 4:30 and 5 p.m. in 3-115. Due on Thursday, December 18th by 5 p.m. in 3-115.

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.

The Instructor may be reached by e-mail at kamenova@math.sunysb.edu.

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