Module description
Aims:
The main aim of the module is to introduce students to basic concepts from abstract algebra, especially the notion of a group. The module will help prepare students tor further study in abstract algebra as well as familiarize them with tools essential in many other areas of mathematics. The module is also intended to help students in the transitions from concrete to abstract mathematical thinking and from a purely descriptive view of mathematics to one of definition and deduction.
Learning Outcomes:
Understand and be able to use the fundamental theorem of arithmetic, the Euclidean algorithm and modular arithmetic. Understand how arithmetic operations are generalised in the framework of group theory and ring theory. Be able to apply the definitions and basic properties of groups and rings to study simple examples.
Syllabus:
The integers: well-ordering principle, division and the Euclidean algorithm. Prime numbers and the fundamental theorem of arithmetic. Binary operations, equivalence relations and quotient sets. Modular arithmetic. Groups: definition and examples, abelian and cyclic groups, orders of group elements, cosets and Lagrange's theorem. Rings: definitions and examples, unit groups. The Chinese remainder theorem. Polynomials.
Teaching arrangements
Three hours of lectures each week, and a weekly tutorial of one hour.
Prerequisites
None
Formative assessment
Problem sheets are given out at the end of each week. You are expected to hand in your work on these sheets the following week so that this can be marked, returned and discussed at the subsequent tutorial
Required reading/resources
None
Suggested reading/resources (link to My Reading Lists)
Assessment details
Written examination
Class tests
Educational aims & objectives
The main aim of the module is to introduce students to basic concepts from abstract algebra, especially the notion of a group. The module will help prepare students tor further study in abstract algebra as well as familiarize them with tools essential in many other areas of mathematics. The module is also intended to help students in the transitions from concrete to abstract mathematical thinking and from a purely descriptive view of mathematics to one of definition and deduction.
Learning outcomes
Understand and be able to use the fundamental theorem of arithmetic, the Euclidean algorithm and modular arithmetic. Understand how arithmetic operations are generalised in the framework of group theory and ring theory. Be able to apply the definitions and basic properties of groups and rings to study simple examples.
Teaching pattern
Three hours of lectures each week, and a weekly tutorial of one hour.