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MATH3031: NUMBER THEORY III

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Type Open
Level 3
Credits 20
Availability Available in 2023/24
Module Cap
Location Durham
Department Mathematical Sciences

Prerequisites

  • Algebra II (MATH2581).

Corequisites

  • None.

Excluded Combinations of Modules

Aims

  • To provide an introduction to Algebraic Number Theory (DiophantineEquations 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 solvenovel and/or complex problems in Number Theory.
  • have a systematic and coherent understanding of theoreticalmathematics in the field of Number Theory.
  • have acquired a coherent body of knowledge of these subjectsdemonstrated 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 mathematicalskills 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 theapplication of the theory to practical examples.
  • Assignments for self-study develop problem-solving skills andenable students to test and develop their knowledge and understanding.
  • Formatively assessed assignments provide practice in theapplication of logic and high level of rigour as well as feedback forthe students and the lecturer on students' progress.
  • The end-of-year examination assesses the knowledge acquiredand the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

ActivityNumberFrequencyDurationTotalMonitored
Lectures422 per week for 20 weeks and 2 in term 31 Hour42 
Problems Classes8Four in each of terms 1 and 21 Hour8 
Preparation and Reading150 
Total200 

Summative Assessment

Component: ExaminationComponent Weighting: 100%
ElementLength / DurationElement WeightingResit Opportunity
Written examination3 hours100none

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.

More information

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