Course Code:        MATH 3272

Course Title:         Abstract Algebra II

Course Type:        Core

Level:                    3

Semester:             1

No. of Credits:       3

Pre-requisite(s):     (MATH 2272) and (MATH 2273)

 

 

Course Rationale  

 

Mathematics is an engaging field that provides powerful tools for all fields of applied science. The structures that one meets in the study of abstract algebra are ubiquitous in all areas of Mathematics and in-depth knowledge of the same should be considered fundamental for students and practitioners alike. The central ideas and proofs emerging from this field have shaped human thought over the years. Those who have contributed to the development of this field are considered among the greatest thinkers of all times, and their achievements are held in high esteem.

 

This course serves to sharpen the analytical and critical reasoning skills of the student, and to improve his/her ability to express sophisticated mathematical ideas with clarity and coherence. An important objective of the course is to provide students with an opportunity to enhance their ability to abstract ideas, as this is one of the defining characteristics of any good mathematician. Particular emphasis will be placed on the powerful techniques and results that one obtains by investigating the structure of several mathematical objects in their full generality. 

 

Students taking this course will be drilled in the practice of analytical reasoning, in order to obtain solutions and to interpret results. For this reason, this course is particularly relevant for students who intend to teach mathematics at the high school level. As the process of mathematical abstraction is central to the development of a mathematician, this is a core course for students pursuing the major in Mathematics.

 

 

Course Description        

 

Students who take this course will require knowledge of the basic concepts of Algebra. Thus, ABSTRACT ALGEBRA I and LINEAR ALGEBRA are both listed as pre-requisites.

 

The first part of the course continues the treatment of Groups started in ABSTRACT ALGEBRA I. Some important subgroups are defined, and the important concept of a group acting on a set is introduced. The power of group actions is demonstrated by using the technique to prove several key results about finite groups. The investigation of finite groups is concluded with the famous Sylow Theorems.

 

The construction of the (finite) direct product should be familiar to any mathematician, and so the course proceeds to do this. Abelian groups are discussed briefly; a statement of the Decomposition Theorem for finite groups is given. The section on Group Theory is concluded with a discussion of subgroup series – an important technique in determining the structure of a group. The Jordan-Holder Theorem is proved, and an important class of groups - the solvable groups are introduced.

 

The course then shifts focus to one of the most important examples of a Euclidean ring – the polynomial ring over a field. (Euclidean rings were introduced in ABSTRACT ALGEBRA I.) The fundamental results that transfer from Euclidean rings are restated in context, and the idea of irreducibility is introduced. The course then specialises to the rational field, and several key results concerning polynomials over the rationals are proved.

 

The course naturally progresses to investigate the existence of roots of polynomials over their base field. The extremely important construction of the algebraic extension containing the root of a polynomial is done in detail, with several interesting and motivating examples. The course continues to prove the existence of a splitting field, and concludes with a statement of the Fundamental Theorem of Algebra. Straightedge and compass constructions will be presented as an application if time permits.

 

Since cogent communication of mathematical ideas is important in the presentation of proofs, the course will emphasize clear, concise exposition. This course will therefore be useful for all students who wish to improve their skills in mathematical proof and exposition, or who intend to study more advanced topics in mathematics.

 

Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through marked assignments supplemented bu (unmarked) problem papers, allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures. Assessment will be based on the marked assignments and in-course tests followed by a final examination based on the whole course

 

Learning Outcomes       

 

Upon successful completion of the course, students will be able to:

 

     • Construct new groups from existing (simpler) groups

  • Manipulate groups of permutations
  • Use group actions to solve non-trivial problems
  • Define p-groups and appreciate their fundamental importance.
  • State and prove the Sylow Theorems
  • Define composition series and prove the Jordan-Holder Theorem
  • Define solvable groups in terms of solvable series
  • Manipulate rings and ideals
  • Give examples of rings that occur naturally in mathematics
  • State the formal definition of a polynomial
  • Manipulate and factorise polynomials over the rational field
  • Construct and investigate algebraic extensions of fields
  • Prove the existence of splitting fields
  • Formulate non-trivial problems in an algebraic context

 

 

Course Content      

 

Conjugates and commutators and related subgroups. Group actions. Finite groups Direct Products. Abelian groups. Subgroup series. Polynomial Rings. Algebraic extensions. Splitting fields.

 

 

Teaching Methodology

 

Lectures:                 Three lectures each week (50 minute each).

 

 

Assessment

                                                                                                                                                         

Coursework - Two coursework exams (20% each), Assignments (10%).  Total: 50%

FinalExamination- One 2-hour written paper.                                                                                 Total: 50%

 

 

Course Calendar

 

WEEK

TOPICS

TUTORIAL & CW EXAM DISCUSSED

1

Course Overview/Introduction

 

Review of Group Theory. Conjugates and commutators.

-

2

Group actions

 

3

Finite groups. The Sylow Theorems.

Assignment 1 due.

4

Direct products and abelian groups.

 

5

Composition series. The Jordan-Holder Theorem.

Assignment 2 due.

6

Solvable groups.

 

7

Polynomial rings over an arbitrary field.

Assignment 3 due.

8

Polynomials over the rationals.

1st coursework examination

9

Integers, rationals

Assignment 4 due.

10

Review of field extensions. Algebraic extensions.

 

11

Splitting fields.

Assignment 5 due.

12

Straightedge and compass constructions.

2nd coursework examination

13

Revision

Assignment 6 due.

                                 

Required Reading

 

Essential Texts:

 

  • Joseph J. Rotman. A First Course In Abstract Algebra. Pearson, Third edition, 2005
  • I.N. Herstein. Topics in Algebra. John Wiley & Sons, Second edition, 1976
  • David S. Dummit and Richard M. Foote. Abstract Algebra. John Wiley & Sons, Third edition, 2004

 

 

Other Reference Texts:

Introduction to Abstract Algebra by Neal McCoy , Trustworthy Communications, 7th edition, ISBN- 0982263317, (2009).

Modern  Abstract Algebra – Frank Ayres, Mc Graw -Hill, , ISBN 978007002651, (2009).