Page, in Encyclopedia of Physical Science and Technology (Third Edition), 2003 I.F Prime NumbersĪ subject of early investigation in number theory and one of continuing interest today is that of prime numbers. Williams, Introductory Algebraic Number Theory, Cambridge University Press, 2003 Raw scores are moderated as described in the Undergraduate Handbook. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores on the examinations will be determined according to the marking scheme written on the examination paper.
#Basic number theory concepts how to
Using an abstract framework to better understand how to attack a concrete problem. learned how to assimilate material from several sources into a coherent document.gained an appreciation of how the basic ideas may be further developed.developed an awareness of a broader literature.Solve certain Diophantine equations by applying tools from the course.īy pursuing an individual project on a more advanced topic students should have:.Find the factorisation of ideals, the ring of integers, the class number and ideal class group of a number field.Define, describe and analyse more advanced concepts such as ideals, ideal classes, unit groups, norms, traces and discriminant.Understand and clearly define number fields and their ring of integers, in particular quadratic number fields and cyclotomic number fields.Students who successfully complete the unit should be able to: Students may not take this unit with the corresponding Level 6 unit MATH36205 Algebraic Number Theory 3. The material is complementary to that of MATHM0007 Analytic Number Theory.
The course build on the material of MATH21800 Algebra 2 and has relations to MATHM2700 Galois Theory. By the end of the units, all these tools will be used to solve various Diophantine equations. We will see that unique factorisation doesn't work in number fields and therefore we will introduce ideals (an analogue of numbers) to go around that problem.
In this course we will focus on number fields (extensions of the rationals), their ring of integers (the analogue of the integers) and their various properties. So when we come across an equation, say for example the one that arises from Pythagoras Theorem, we can be tempted to ask: which integers solves these equations, and can we find all of them? Trying to find all integer solutions to a given equation is called solving Diophantine equations, and Number Theory is the study of Diophantine equations.īroadly speaking Algebraic Number Theory tries to solve number theory questions by using tools and techniques from abstract algebra. Integers and rational numbers are the first numbers we encounter, and as such they are, in a way the easiest numbers to think with. To become comfortable in using tools and techniques from algebraic number theory to solve Diophantine equations.To become familiar with concepts such as number fields, rings of integers and ideals.To gain an understanding and appreciation of algebraic number theory.MATHM2700 Galois Theory is recommended but not necessary. MATH30200 Number Theory, and MATH33300 Group Theory are recommended but not necessary. Please see the current academic year for up to date information. Please note: you are viewing unit and programme informationįor a past academic year.