Course Code:                 MATH 2272

Course Title:                  Abstract Algebra I

Level:                             2

Semester:                       2           

No. of Credits :                3

Pre-requisite(s):               MATH 1141 and MATH 1152

 

Course Rationale

 This course serves to sharpen the analytical and critical reasoning skills of the student, and to improve his/her ability to express mathematical ideas with clarity and coherence.  An important objective of this course is to provide students with an opportunity to enhance their ability to reason, as this is one of the defining characteristics of a mathematician.  Particular emphasis will be placed on the statement of abstract ideas and on the process of refining these ideas to make them more tractable whenever necessary.  This course will be useful to those students who want to move on to more advanced topics, as well as those who want to gain a depth of understanding of familiar algebraic structures.

 

Course Description

Students who take this course will require a basic grounding in set theory and logic.  For this reason, Math 1152 is listed as a prerequisite.

This course introduces students to basic structures of abstract algebra, including groups, rings and fields.  In the introduction, the focus is on binary operations and equivalence relations, which will be used throughout this course.  Then groups are introduced, and students will learn that they come in many varieties.  Subgroups and maps between groups are studied.  In the second part of the course, rings are studied.  Again, examples are studied, some familiar and some new.  As usual, subrings, ideals and maps between rings are studied.  After this, Euclidean rings are studied.  Finally, a brief introduction to fields is given.

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

By the end of the course, students will be able to:

  • Define equivalence relations and binary operations.
  • Define the group and ring axioms.
  • Give basic examples and non-examples of cyclic, finite and abelian groups.
  • Determine whether a given subset of a group is a subgroup, and whether a given subset of a ring is a subring, or an ideal.
  • Determine whether a given subgroup is a normal subgroup.
  • Explain the definition of the quotient group G/H, and prove that it is a group.
  • Explain the definition of the quotient ring R/I, and prove that it is a ring.
  • Define a group and ring homomorphism and apply the isomorphism theorems.
  • Define the symmetric and alternating groups, find the cycle decomposition of a permutation, find the parity of a permutation.
  • Prove that a Euclidean ring is a principal ideal ring.
  • Define division ring and field, and give examples of both.
  • Explain the field of quotients of an integral domain.
  • Perform arithmetic in a finite field.

Content                     

1. Sets and Relations.

Equivalence relations, binary operations.

2. The Definition of a Group.

The definition of a group.  Examples of a group: numbers, symmetries, matrices.  Properties of groups: cyclic, abelian, finite.

3. Subgroups, Quotient Groups and Group Homomorphisms.

Subgroups. Cosets and Lagrange’s Theorem.  The Euler-Fermat Theorem as a consequence of Lagrange’s Theorem.  Wilson’s Theorem.  Normal subgroups.  The construction of a quotient group.  Generating sets.  Homomorphisms of groups.  The kernel of a homomorphism.  The isomorphism theorems.

4. Permutation Groups.

The symmetric group.  Transpositions and cycles.  Cycle decomposition and cycle structure.  The alternating group.

5. The Definition of a Ring.

The definition of a ring.  Examples of rings.  Special classes of rings.  Associativity and commutativity.  Zero-divisors and integral domains.

6. Ideals, Quotient Rings, and Ring Homomorphisms.

One-sided and two-sided ideals.  The construction of the quotient ring.  Maximal ideals. Principal ideals.  Prime ideals.  Homomorphisms of rings.  The ring isomorphism theorems.

7. Euclidean Rings.

The defining properties of Euclidean rings.  Euclidean rings as principal ideal rings.  Divisibility and primality.

8. Division Rings.

Properties of division rings.  The definition of a field.  The construction of the field of quotients of an integral domain.  The characteristic of a field.  Subfields and field extensions.  The prime subfield.  The integers modulo a prime, p.

 

Teaching Methodology

Lectures: Three lectures odd weeks, two lectures on even weeks (50 minutes each).

Tutorial: One tutorial session every alternate week (50 minutes each).

 

Assessment

Final exam:  2-hour written paper                              50%

Two Midterm Exams (40%), Assignments (10%)        50%.

 

 

Week

Lecture subjects

Assignments

Tutorial

1

Course Overview/Introduction

 

Equivalence relations, binary operations.  The definition of a group.  Examples of a group: numbers, symmetries, matrices.

Assignment 1 given

 

2

Properties of groups: cyclic, abelian, finite.

 

Tutorial 1

3

Subgroups.  Cosets and Lagrange’s Theorem.  The Euler-Fermat Theorem as a consequence of Lagrange’s Theorem.  Wilson’s Theorem.

Assignment 1 due,

Assignment 2 given

 

4

Normal subgroups.  The construction of a quotient group.  Generating sets. 

 

Tutorial 2

5

Homomorphisms of groups.  The kernel of a homomorphism.  The isomorphism theorems.  The symmetric group.  Transpositions and cycles.

Assignment 2 due,

Assignment 3 given

 

6

Cycle decomposition and cycle structure.  The alternating group.

 

Exam 1

7

The definition of a ring.  Examples of rings.  Special classes of rings.  Associativity and commutativity.  Zero-divisors and integral domains.

Assignment 3 due,

Assignment 4 given

 

8

One-sided and two-sided ideals.  The construction of the quotient ring.

 

Tutorial 3

9

Maximal ideals. Principal ideals.  Prime ideals.  Homomorphisms of rings.  The ring isomorphism theorems.

Assignment 4 due,

Assignment 5 given

 

10

The defining properties of Euclidean rings.  Euclidean rings as principal ideal rings.  Divisibility and primality.

 

Tutorial 4

11

Properties of division rings.  The definition of a field.  The construction of the field of quotients of an integral domain.

Assignment 5 due

 

12

The characteristic of a field.  Subfields and field extensions.  The prime subfield.  The integers modulo a prime, p.

 

Exam 2

13

Revision

 

 

 

 

Reference Material

Recommended

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.