Undergraduate Programme and Module Handbook 2005-2006 (archived)

Module MATH2061: ALGEBRA AND NUMBER THEORY II

Department: MATHEMATICAL SCIENCES

MATH2061: ALGEBRA AND NUMBER THEORY II

Type	Open	Level	2	Credits	20	Availability	Available in 2005/06	Module Cap	None.	Location	Durham

Prerequisites

Core Mathematics A (MATH1012)

Corequisites

Linear Algebra II (MATH2021).

Excluded Combination of Modules

Mathematics for Engineers and Scientists (MATH1551), Single Mathematics A (MATH1561), Single Mathematics B (MATH1571), Foundation Mathematics (MATH1641)

Aims

To introduce further concepts in abstract algebra, develop their theory, and apply them to solve problems in number theory and other areas.

Content

Examples of groups.
Group actions.
Homomorphisms and quotient groups.
Finitely generated abelian groups.
Rings and fields.

Learning Outcomes

Subject-specific Knowledge:

By the end of the module students will: be able to solve a range of predictable and unpredictable problems on Algebra and Number Theory.
have an awareness of the abstract concepts of theoretical mathematics in the field of Algebra and Number Theory.
have a knowledge and understanding of fundamental theories of these subjects demonstrated through one or more of the following topic areas: Example of groups, generators, homomorphisms.
Conjugacy, centre.
Group actions, Equivalence relations.
Cauchy's Theorem, Burnside's Theorem.
Quotient groups.
First Isomorphism Theorem.
Structure of finitely generated abelian groups.
Rings.
Polynomial ring is one variable over a field.

Subject-specific Skills:

In addition students will have the ability to undertake and defend the use of alternative mathematical skills in the following areas with minimal guidance: Abstract reasoning.

Key Skills:

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

Lecturing demonstrates what is required to be learned and the application of the theory to practical examples.
Weekly homework problems provide formative assessment to guide students in the correct development of their knowledge and skills.
Tutorials provide active engagement and feedback to the learning process.
The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Activity	Number	Frequency	Duration	Total/Hours
Lectures	42	2 per week for 19 weeks and 1 in term 3	1 Hour	42
Tutorials	10	Fortnightly for 20 weels	1 Hour	10
Problems Classes	10	Fortnightly for 20 weeks	1 Hour	10
Preparation and Reading				138
Total				200

Summative Assessment

Component: Examination		Component Weighting: 100%
Element	Length / duration	Element Weighting	Resit Opportunity
three hour written examination		100%

Formative Assessment:

One written assignment to be handed in every third lecture in the first 2 terms. Normally each will consist of solving problems from a Problems Sheet and typically will be about 2 pages long. Students will have about one week to complete each assignment.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University