Durham University
Programme and Module Handbook

Undergraduate Programme and Module Handbook 2026-2027

Module MATH3031: Number Theory III

Department: Mathematical Sciences

MATH3031: Number Theory III

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

Prerequisites

  • Algebra II (MATH2781 or MATH2581).

Corequisites

  • None.

Excluded Combination of Modules

    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: Examination Component Weighting: 70%
    Element Length / duration Element Weighting Resit Opportunity
    On Campus Written Examination 2 hours 100%
    Component: Summative Assignments Component Weighting: 30%
    Element Length / duration Element Weighting Resit Opportunity
    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.