New York City College of Technology The City University of New York DEPARTMENT: Mathematics PREPARED BY: Professor Andrew Douglas, Professor Delaram Kahrobaei COURSE: MAT 3080 TITLE: Modern Algebra DESCRIPTION: An introductory course in modern algebra covering groups, rings and fields. Topics in group theory include permutation groups, cyclic groups, dihedral groups, subgroups, cosets, symmetry groups and rotation groups. In ring and field theories topics include integral domains, polynomial rings, the factorization of polynomials, and abstract vector spaces. TEXT: Gallian, J.A. (2010). Contemporary Abstract Algebra, 7th Ed. Brooks/Cole Cengage Learning. CREDITS HOURS: 3 cl hrs, 0 lab hrs, 3 cr PREREQUISITES: MAT 2580, MAT 3075 LEARNING OBJECTIVES: For successful completion of the course, students should be able to: 1. Define the terms group, ring and field and be able to give examples of each of these kinds of algebraic structures. 2. Define terms (such as homomorphism, subgroup and integral domain) and state theorems (such as Lagrange’s Theorem) of modern algebra. 3. Apply concepts, terminology and theorems to solve problems and prove simple propositions in modern algebra. 4. Describe applications and relationships of group theory to geometry.

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INSTRUCTIONAL OBJECTIVES

ASSESSMENT

For successful completion of the course, students should be able to: Define the terms group, ring and field and be able to give examples of each of these kinds of algebraic structures. Define the concept of a subgroup and determine (prove or disprove), in specific examples, whether a given subset of a group is a subgroup of the group. Solve problems and prove simple propositions involving concepts, terms and theorems of group theory. Compare rings, fields and integral domains.

Instructional Activity, Evaluation Methods and Criteria Class and Blackboard discussion, Tests, Final Exam.

Solve problems and prove simple propositions involving concepts, terms and theorems of ring theory. Apply the reducibility and the irreducibility tests for polynomials. Describe applications and relationships of group theory to geometry

Graded Homework, Group Work, Tests, Final Exam. Graded Homework, Group Work, Tests, Exam. Class and Blackboard Discussion, Graded Homework. Graded Homework, Group Work, Tests, Exam. Graded Homework, Group Work, Tests, Exam. Graded Homework, Class and Blackboard discussion, Tests, Exam.

GRADING PROCEDURE: • • • • •

Homework Assignments In Class Tests Final Exam Projects Presentations

20% 10% Each (3 tests) 35% 10% 5%

TEACHING/LEARNING METHODS: • • • • •

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Lecture and guided discussion Blackboard discussion Homework assignments Group project and group work Technology: A computer algebra system such as MAPLE will be used to facilitate the exploration of mathematical concepts.

WEEKLY COURSE OUTLINE: WEEK 1 2 3 4 5 6-8

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TOPIC Preliminaries: Properties of Integers, Modular Arithmetic, Mathematical Induction, Equivalence Relations. Motivation and Introduction: Symmetries of a Square, the Dihedral Group, applications of the Dihedral group (e.g., Designing a Zip code reader) Groups: Definition, Examples, Elementary Properties of Groups. Subgroups: Terminology and Notation, Subgroup Test, Examples. Cyclic Groups and Permutation Groups: Definitions and Basic Properties. Normal Subgroups: Definitions, examples, applications. Homomorphism: Definitions, examples, properties. Cosets and Lagrange’s Theorem: Properties of Cosets, Lagrange’s Theorem, the Rotation Group of a Cube. Symmetry Groups: Isometries, Finite Plane Symmetry Groups, Finite Groups of Rotation in R3, the groups of rotation of the platonic solids, the Euclidean Group in R2 and R3. Introduction to Rings: Definitions and Motivation, Examples of Rings, Properties of Rings, Subrings. Integral Domains: Definition and Examples, Fields. Polynomial Rings: Notation and Terminology, The Division Algorithm and Consequences. Factorization of Polynomials: Reducibility Tests, Irreducibility Tests, Factorization in Z[x]. Vector Spaces: Definitions and examples of vector spaces, subspaces, linear independence. Final Exam

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