Undergraduate Programme and Module Handbook 2024-2025

# Module MATH3031: NUMBER THEORY III

## Department: Mathematical Sciences

### MATH3031: NUMBER THEORY III

Type | Open | Level | 3 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | Location | Durham |
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#### Prerequisites

- Algebra II (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.
- 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 assignments to be submitted.

■ 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