Module MATH4211: Number Theory IV

Department: Mathematical Sciences

MATH4211: Number Theory IV

Prerequisites

• Mathematics modules to the value of 100 credits in Years 2 and 3, with at least 40 credits at Level 3, and including Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061).

Corequisites

• None

Excluded Combination of Modules

• Number Theory III (MATH3031)

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.
• Fields.
• Algebraic integers.
• Quadratic fields and integers.
• The discriminant and integral bases.
• Factorization of ideals.
• The ideal class group.
• Units in quadratic fields.
• Reading material on a topic in one of the above areas for fields of degree greater than 2.

Learning Outcomes

• 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.
• Student will also have a knowledge and understanding of a topic in the areas listed under content, for fields of degree greater than 2.

• In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Abstract reasoning.
• Students will have an ability to read independently to acquire knowledge and understanding in the area of higher degree fields.

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.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding in the area of higher degree fields.
• 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. enter text as appropriate for the module

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four 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