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MTH211Sciences3 Unitsintermediate

Introductory Set Theory And Abstract Algebra

This course introduces students to Set Theory and Abstract Algebra, providing a foundation for advanced studies in mathematics, computer science, and communications technology. It covers fundamental algebraic concepts, including sets, functions, groups, subgroups, polynomial rings, integral domains, and field extensions. Students will learn to solve problems related to these topics and develop rigorous analytical skills. The course aims to prepare students for more advanced courses in algebra.

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208h
Study Time
13
Weeks
16h
Per Week
intermediate
Math Level
Course Keywords
Set TheoryAbstract AlgebraGroup TheoryPolynomial RingsField Extensions

Course Overview

Everything you need to know about this course

Course Difficulty

Intermediate Level
Builds on foundational knowledge
65%
intermediate
📊
Math Level
Moderate Math
📖
Learning Type
Theoretical Focus

Course Topics

Key areas covered in this course

1

Sets and Functions

2

Group Theory

3

Subgroups

4

Lagrange's Theorem

5

Integral Domains

6

Field Theory

7

Polynomial Rings

8

Irreducibility

9

Field Extensions

Total Topics9 topics

Requirements

Knowledge and skills recommended for success

MTH131 - Elementary Set Theory

💡 Don't have all requirements? Don't worry! Many students successfully complete this course with basic preparation and dedication.

Assessment Methods

How your progress will be evaluated (3 methods)

Assignments

Comprehensive evaluation of course material understanding

Written Assessment

Tutor-Marked Assessments

Comprehensive evaluation of course material understanding

Written Assessment

Final Examination

Comprehensive evaluation of course material understanding

Computer Based Test

Career Opportunities

Explore the career paths this course opens up for you

Data Analyst

Apply your skills in this growing field

Cryptography

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Software Developer

Apply your skills in this growing field

Theoretical Physicist

Apply your skills in this growing field

Mathematics Teacher

Apply your skills in this growing field

Industry Applications

Real-world sectors where you can apply your knowledge

CryptographyCoding TheoryData ScienceTelecommunicationsQuantum Computing

Study Schedule Beta

A structured 13-week journey through the course content

Week
1

Module 1: Introduction

3h

Unit 1: Sets and Functions

3 study hours
  • •Review definitions of sets, subsets, and set operations.
  • •Practice problems involving unions, intersections, and complements of sets.
  • •Solve problems related to Cartesian products and relations.
Week
2

Module 1: Introduction

3h

Unit 1: Sets and Functions

3 study hours
  • •Define and identify reflexive, symmetric, and transitive relations.
  • •Determine equivalence classes for given equivalence relations.
  • •Practice problems involving different types of functions: one-to-one, onto, and bijective.
Week
3

Module 1: Introduction

3h

Unit 2: Groups

3 study hours
  • •Define binary operations and determine if they are commutative or associative.
  • •Identify identity elements and inverses for given binary operations.
  • •Practice problems involving groups, subgroups, and abelian groups.
Week
4

Module 1: Introduction

3h

Unit 2: Groups

3 study hours
  • •Apply cancellation laws and laws of indices to solve group-related problems.
  • •Study the properties of integers modulo n and solve related problems.
  • •Explore the symmetric group and its properties.
Week
5

Module 1: Introduction

3h

Unit 3: Subgroups

3 study hours
  • •Define subgroups and check if a subset of a given group is a subgroup.
  • •Explore properties of subgroups, including intersection and union.
  • •Solve problems related to cyclic groups and their generators.
Week
6

Module 1: Introduction

3h

Unit 4: Lagrange's Theorem

3 study hours
  • •Form left and right cosets of a subgroup.
  • •Partition a group into disjoint cosets of a subgroup.
  • •Apply Lagrange's Theorem to solve problems related to group and subgroup orders.
Week
7

Module 2: Advanced Topics

3h

Unit 1: The Basics

3 study hours
  • •Define integral domains and check if an algebraic system is an integral domain.
  • •Obtain the characteristic of any ring.
  • •Check whether an algebraic system is a field or not.
Week
8

Module 2: Advanced Topics

3h

Unit 1: The Basics

3 study hours
  • •Define and identify prime ideals and maximal ideals.
  • •Prove and use simple properties of integral domains and fields.
  • •Construct or identify the field of quotients of an integral domain.
Week
9

Module 2: Advanced Topics

3h

Unit 2: Polynomial Rings

3 study hours
  • •Identify polynomials over a given ring.
  • •Prove and use the fact that R[x], the set of polynomials over a ring R, is a ring.
  • •Relate certain properties of R[x] to those of R.
Week
10

Module 2: Advanced Topics

3h

Unit 2: Polynomial Rings

3 study hours
  • •Prove and use the division algorithm for F[x], where F is a field.
  • •Solve problems related to roots of polynomials.
  • •Apply the Remainder Theorem to find remainders of polynomial divisions.
Week
11

Module 2: Advanced Topics

3h

Unit 3: Special Integral Domains

3 study hours
  • •Check whether a function is a Euclidean valuation or not.
  • •Identify principal ideal domains.
  • •Identify unique factorization domains.
Week
12

Module 2: Advanced Topics

3h

Unit 3: Special Integral Domains

3 study hours
  • •Obtain the g.c.d of any pair of elements in a unique factorization domain.
  • •Prove and use the relationship between Euclidean domains, principal ideal domains, and unique factorization domains.
Week
13

Module 2: Advanced Topics

3h

Unit 4: Irreducibility and Field Extensions

3 study hours
  • •Prove and use Eisenstein's criterion for irreducibility in Q[x].
  • •Obtain field extensions of a field F from F[x].
  • •Obtain the prime field of any field.

This study schedule is in beta and may not be accurate. Please use it as a guide and consult the course outline for the most accurate information.

Course PDF Material

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Study Tips & Exam Preparation

Expert tips to help you succeed in this course

1

Create detailed concept maps linking key definitions and theorems from Modules 1 and 2.

2

Practice solving a variety of problems related to groups, subgroups, and cosets from Units 2-4.

3

Focus on understanding and applying Lagrange's Theorem to determine possible subgroup orders.

4

Master the division algorithm for polynomials and practice finding quotients and remainders.

5

Review the definitions and properties of integral domains, fields, and their characteristics.

6

Study Eisenstein's criterion and practice applying it to determine irreducibility of polynomials.

7

Work through all examples and self-assessment exercises in the course materials.

8

Allocate specific time slots for focused study and problem-solving each week.

9

Form a study group with fellow students to discuss challenging concepts and share insights.

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