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Durham University

Faculty Handbook Archive

Archive Module Description

This page is for the academic year 2019-20. The current handbook year is 2020-21

Department: Mathematical Sciences

MATH3031: NUMBER THEORY III

Type Open Level 3 Credits 20 Availability Available in 2019/20 Module Cap Location Durham

Prerequisites

  • Algebra II (MATH2581).

Corequisites

  • None.

Excluded Combination of Modules

  • Number Theory IV (MATH4211).

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.
  • Dirichlet's Unit Theorem.
  • L-functions.
  • Class number formula for quadratic fields.

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 specialised mathematical skills in the following areas which can be used with minimal guidance: Abstract reasoning.
Key Skills:

    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.
    • Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
    • Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
    • 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 20 weeks and 2 in term 3 1 Hour 42
    Problems Classes 8 Four in each of terms 1 and 2 1 Hour 8
    Preparation and Reading 150
    Total 200

    Summative Assessment

    Component: Examination Component Weighting: 100%
    Element Length / duration Element Weighting Resit Opportunity
    Written examination 3 hours 100% none

    Formative Assessment:

    Eight written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.


    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