MATH41620: Number Theory

• Prior knowledge of Algebra at undergraduate level.

• None

• None

• To provide an introduction to Algebraic Number Theory (Diophantine Equations and Ideal Theory).

• 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.
• Reading material on a topic related to one of the above areas.

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.
• Student will also have a knowledge and understanding of a topic related to the areas listed under content.

Subject-specific Skills:

• 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 related areas.

Key Skills:

• 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 related areas.
• 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.

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

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