Module MATH3031: Number Theory III

Department: Mathematical Sciences

MATH3031: Number Theory III

Type	Level	Credits	Availability	Module Cap	Location
Open	3	20	Available in 2026/2027		Durham

Pre-requisites

Algebra II (MATH2781 or MATH2581).

Corequisites

None.

Excluded Combinations

Aims

To provide an introduction to Algebraic Number Theory (Diophantine Equations and Ideal Theory).

Content

Diophantine equations using elementary methods.
Unique factorization.
Ideals.
Euclidean rings.
Number fields.
Algebraic integers.
Quadratic fields and integers.
The discriminant and integral bases.
Factorization of ideals.
The ideal class group.

Learning Outcomes

Subject-specific Knowledge

By the end of the module students will:
be able to solve novel and/or complex problems in Number Theory.
have a systematic and coherent understanding of theoretical mathematics in the field of Number Theory.
have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Euclidean rings, principal ideal domains, uniqueness of factorization. Algebraic number fields (especially Quadratic fields). Applications to Diophantine equations.

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

Students will have basic mathematical skills in the following areas: abstract reasoning, problem solving.

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

Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
Summative assignments test achievement of learning outcomes and provide feedback to students about their mastery of the topics.
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	Attendance Monitored
Lectures	40	2 per week for 20 weeks	1 Hour	40
Problem Classes	10	4 in Michaelmas; 4 in Epiphany; 2 in Easter	1 Hour	10	Yes
Preparation and Reading				150
Total				200

Summative Assessment

Component	Component Weighting	Element	Length/Duration	Element Weighting	Resit Opportunity
Examination	70%	On Campus Written Examination	2 hours	100%
Summative Assignments	30%	Assignment		100%

Formative Assessment

Four assignments to be submitted.

Students who do not attend monitored activities shown under Teaching Methods and Learning Hours, or who fail to complete the summative or formative assessment(s) specified above, may be subject to the Academic Progress procedures defined in the University's General Regulation V, and may be required to leave the University.