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Announcement: The final exam will be held on December 22, Thursday 8am-10:30am in our usual classroom. See program for the exam.

Classes: TuTh 9:50am-11:10am Harriman Hall 115 Instructor: Valentina Kirichenko (office Math 3-102, phone 632-8884) Webpage: http://www.math.sunysb.edu/~vkiritch/ Office Hours: Tu 3-5, Th 2-3 (Math 3-102), Th 3-4 (MLC) and by appointment

Grader: Wenchuan Hu Webpage: http://www.math.sunysb.edu/~wenchuan/ Office Hours:

About this Course: 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 reading:

• 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.

Homework/Class Participation 30%, Midterm Test 30%, Final Exam 40%

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.

August 30, September 1
Sections 0.1-0.3, 1.1, 1.4; Examples of groups: the group of integers, the group of integers modulo n, the multiplicative group of invertible integers modulo n. Euclidean algorithm for the greatest common divisor of two numbers. Definition of group, matrix groups.
Homework 1 (due September 8)

September 6, 8
Sections 1.2, 1.3, 1.6; Homomorphisms, isomorphisms, normal subgroups. Further examples of groups: groups of permutations, dihedral groups.
Homework 2 (due September 15)

September 13, 15
Sections 1.7, 2.1-2.3, 3.1, 3.2, 4.1; Subroups, cosets, quotient groups and Lagrange's theorem. More about symmetric groups (aka groups of permutations). Group actions: orbits, stabilizers.
Homework 3 (due September 22)
The problem marked by an asterisk is difficult.

September 20, 22
Sections 3.5, 4.2, 4.3; Group actions: actions by left multiplication and by conjugation. Sign of a permutation, alternating group.
Homework 4 (due September 29)

September 27, 29
Sections 4.4, 5.1, 5.2; Automorphisms and inner automorphisms. Direct products. Classification of Abelian groups.
Homework 5 (due October 6)

October 6
Sections 5.2, 6.1; Proof of the classification theorem for Abelian groups.
Homework 6 (due October 18 (Tuesday))

October 11
For additional reference see Chapter 12, Section 4 of ``Algebra" by Michael Artin, Prentice Hall, 1991; Elementary row and column operations, the end of the proof of the classification theorem for Abelian groups.
Homework 7 (due October 27)

October 18, 20
Sections 4.5, 5.5, 6.1; Solvable groups, Sylow's theorem, semidirect products.
No homework. Prepare for the midterm.

October 25, 27
Sections 11.1-11.2; Vector spaces, linear operators, bases, coordinates and matrices.

November 1
Section 12.3; see also the article Down with determinants by Sheldon Axler (do not take the title of this excellent article too seriously); Kernel and image of a linear operator, eigenvalues, eigenvectors and Jordan canonical form.
Homework 8 (due November 10)

November 8, 10
Exercises of Section 12.3, for additional reference see Chapter 4, Sections 6--8 of ``Algebra" by Michael Artin; Applications of Jordan canonical form to differential and difference equations. Matrix-valued functions (square root, exponential function etc).
Homework 9 (due November 17)

November 15, 17
Sections 7.1-7.3; Rings, fields, algebras: definitions and examples. Ring homomorphisms. Ideals.
Homework 10 (due November 29 (Tuesday))

November 22
Sections 7.4; Quotient rings. Prime and maximal ideals. Zero divisors and integral domains. Further examples of rings: polynomial rings and rings of formal power series.
Homework 11 (due December 6)

November 29
Sections 7.6, 8.1, 8.2; Chinese Remainder Theorem. Euclidean and Principal Ideal Domains.
Homework 12 (due December 13)

December 6, 8
Sections 8.3, 9.3, 9.4; Unique Factorization Domains. Factorization in the Gaussian integers. Polynomial rings that are Unique Factorization Domains.