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Online course materials for MATH42132Algebraic Number Theory
Unit code:  MATH42132 
Credit Rating:  15 
Unit level:  Level 4 
Teaching period(s):  Semester 2 
Offered by  School of Mathematics 
Available as a free choice unit?:  N 
Requisites
Prerequisite MATH32011  Commutative Algebra (Compulsory)
Aims
To show how tools from algebra can be used to solve problems in number theory.
Overview
Number theory is arguably one of the oldest and most fascinating bran ches of mathematics. This fascination stems from the fact that there are a great many theorems concerning the integers, which are extremely simple to state, but turn out to be rather hard to prove.
The fundamental objects in algebraic number theory are finite field extensions of Q; socalled number fields. To a number field k one associates a ring O_k called its ring of integers. This ring behaves in some respects like the usual ring of integers Z, however many well know properties of Z do not pass over; the most important being that the fundamental theorem of arithmetic can fail in O_k .
The main focus of this course is on the failure of the unique factorisation. We also give a number applications to the study of certain diophantine equations.
Learning outcomes
Learning outcomes. On successful completion of this course unit students will
• Be able to calculate the ring of integers of simple number fields.
• Understand how the class group controls the failure of unique factoristion.
• Be able to calculate the class group in simple examples.
• Be able to factorise ideals into products of prime ideals.
• Solve some nonlinear diophantine equations.
Assessment methods
 Other  10%
 Written exam  90%
Syllabus
Fields and rings [2 lectures]
 Review of required tools from the theory of fields and rings
 Field extensions, ideals, maximal ideals, prime ideals
 Euclidean domain => PID => UFD => integral domain
Number fields [2 lectures]
 Definitions and basic examples
 Embeddings into the real and complex numbers
 Field norms and trace
Rings of integers [4 lectures]
 Integral closures
 Definitions and basic properties
 Discriminants
 Calculation for quadratic field extensions and cyclotomic fields
Unique factorisation of ideals [4 lectures]
 Prime ideals in rings of integers of number fields
 Unique factorisation into prime ideals
Geometry of numbers [4 lectures]
 Lattices
 The Minkowski bound
Failure of unique factorisation [4 lectures]
 Examples
 Definition and finiteness of the class group
Applications [2 lectures]
 Applications to nonlinear Diophantine equations
 Some cases of Fermat’s last theorem
Recommended reading
 Stewart and Tall, Algebraic Number Theory and Fer mat’s Last Theorem.
 Jarvis, Algebraic Number Theory.
 Neukirch, Algebraic Number Theory.
Feedback methods
Feedback tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Coursework or inclass tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer’s office hour.
Study hours
 Lectures  22 hours
 Tutorials  11 hours
 Independent study hours  117 hours
Teaching staff
