Module description
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.
Assessment details
Written examination.
Exercises or quizzes will be set each week to be handed in the following week. These problems will be discussed in the tutorials and solutions will be available.
Semester 1 only students will be set an alternative assessment in lieu of in-person exams in January.
Full year students will complete the standard assessment.
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 and one hour of tutorial per week throughout the term
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists