蘑菇短视频

Skip to main content

MATH 361 Syllabus

Introduction to Abstract Algebra

Course Description

Groups, rings, and fields. Prerequisite: MATH 250. Three credit hours.

Student Learning Objectives

  • Know all relevant definitions and correct statements of major theorems;
  • Explain the defining structures of quotient groups and rings, rings of polynomials and fields;
  • Create examples or counter-examples and apply them appropriately to prove or disprove statements about algebraic structures;
  • Prove statements and solve problems about the structure, size, and nature of groups, subgroups, factor groups, normal groups, rings, ideals, and field using definitions and theorems learned in the course;
  • Incorporate equivalence relations into group theoretic structures, particularly factor groups; create factor groups using normal subgroups or the Isomorphism Theorems and interpret elements of factor groups accurately; and
  • Understand permutations and symmetries in a group theoretic context - particularly the significance of Cayley's Theorem.

Text

I.N. Herstein, Abstract Algebra, third edition, Wiley 1999.

Grading Procedure

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

Course Outline

  • Chapter 1 Things Familiar and Less Familiar
    • Section 1.2 Set Theory
    • Section 1.3 Mappings
    • Section 1.5 The Integers
    • Section 1.6 Mathematical Induction
  • Chapter 2 Groups
    • Section 2.1 Definitions and Examples of Groups
    • Section 2.3 Subgroups
    • Section 2.4 Lagrange's Theorem
    • Section 2.5 Homomorphisms and Normal Subgroups
    • Section 2.6 Factor Groups
    • Section 2.7 The Homomorphism Theorems
    • Section 2.8 Cauchy's Theorem
    • Section 2.9 Direct Products
  • Chapter 3 The Symmetric Group
    • Section 3.1 Preliminaries
    • Section 3.2 Cycle Decomposition
    • Section 3.3 Odd and Even Permutations
  • Chapter 4 Ring Theory
    • Section 4.1 Definitions and Examples
    • Section 4.2 Some Simple Results
    • Section 4.3 Ideals, Homomorphisms, and Quotient Rings
    • Section 4.4 Maximal Ideals
    • Section 4.5 Polynomial Rings
  • Chapter 5 Fields
    • Section 5.1 Examples of Fields

Additional topics selected at the instructor鈥檚 discretion.