MATH42072/MATH32072 - 2009/2010

Unit specification

Aims

The principal aim of this course is to introduce students to the study of the theory of numbers by use of algebraic methods.

Brief description

Algebraic number theory is concerned with the study of the rings generated by the solutions to polynomials with whole number coefficients. By understanding the algebraic properties of such rings one can often gain great insight into the properties of the integers. The factorisation properties of such rings have had some great historic successes: the factorisation properties of the Gaussian integers (i.e. the integer complex numbers) tell one which integers can be represented as the sum of two squares; some cases Fermat’s Last theorem can be dealt with rapidly when there is unique factorisation for the ring generated by the certain roots of unity.
In addition to studying factorisation properties, we shall also study class groups, which measure the extent to which such rings fail to be principal ideal domains; we shall also study the groups of units (or the invertible elements ) of such rings. The intention is then to use these algebraic insights to solve various quadratic and cubic Diophantine (or whole number) equations.

Intended learning outcomes

On successful completion of this course students will have acquired:
An understanding of the notion of rings algebraic integers and their factorisation properties;
The ability to compute class numbers and units of rings of integers of low degree;
The ability to use these techniques to solve certain families of Diophantine equations.

Future topics requiring this course unit

Syllabus

1. Introduction,
2. Dedekind domains,
3. Discriminants, norm and trace,
4. Decomposition of prime ideals,
5. Class numbers and units,
6. Galois action and prime decompostion.

For MATH42072 the lectures will be enhanced by additional reading on further topics.

Textbooks

I.N. Stewart, D.O. Tall, Algebraic Number theory, Chapman and Hall.
A. Frohlich and M Taylor, Algebraic Number theory, Cambridge studies in advances mathematics 27, C.U.P

Assessment

Coursework Test, weighting 20% for MATH32072, 15% for MATH42072,

End of semester examination: two hours weighting 80% for MATH320172, three hours weighting for MATH42072.

 

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