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General

This course is part of the MAGIC core.

Description

We all know how indispensable it is to extend R to C. Often it is better to start with Q and make a finite extension, to get a number field, but there are many more ways to do this-the Galois group of the algebraic closure of Q is immensely complicated. The elements of a number field are algebraic numbers, satisfying monic polynomial equations with coefficients in Q. Among these are the algebraic integers, satisfying monic polynomial equations with coefficients in Z, and they form a subring, the ring of integers. When the number field is Q, this subring is just Z.
This course is about number fields and especially their rings of integers. In general these are not unique factorisation domains, but we shall see how unique factorisation can be restored by using ideals rather than elements. We are led naturally to consideration of the ideal class group and the unit group. This is not just a branch of algebra. We shall use also the geometry of numbers, and some analytic functions, recognising the fact that an algebraic number can be thought of as an element of C, and has a size.

Semester

Spring 2015 (Monday, January 19 to Friday, March 27)

Timetable

• Wed 11:05 - 11:55

Prerequisites

Elementary number theory (primes, linear congruences). Rings and groups, including irreducibles, units, Euclidean domains, quotient groups, First Isomorphism Theorem, and preferably the quotient of a ring by an ideal, and finitely generated abelian groups. Field extensions, including the degree of an extension, irreducible polynomials, and preferably Eisenstein's criterion. Galois theory may be mentioned in passing, or in an exercise, but is not essential.

Syllabus

Finite extensions of Q. Norms, traces and discriminants.
Proof that the subset of algebraic integers is a subring. Existence of an integral basis for the ring of integers.
Ideals, principal ideals, the ideal class group and its finiteness.
Unique factorisation of ideals.
Explicit factorisation of rational primes in rings of integers of number fields.
Minkowski's constant, calculating the ideal class group. Quadratic examples and applications to diophantine equations.
Units. The logarithmic embedding, statement of Dirichlet's unit theorem.
Formula for counting ideals of bounded norm, rough idea of proof. Dedekind zeta function and Dirichlet's class number formula.

Lecturer

 Email J.Manoharmayum@sheffield.ac.uk Phone 0114 2223871

Students

 Robin Green (Manchester) Ahmed HUSSEIN (Leeds) HASSAN IZANLOO (Cardiff) James Macpherson (Sheffield) Jake Massimo (Exeter) Entesar Nasr (Liverpool) Kayvan Nejabati-Zenouz (Exeter) ANON STUDENT (*External) Stephen Worsley (Liverpool) Athirah Zulkarnain (Sheffield)

Bibliography

 Algebraic number theory and Fermat's last theorem Stewart and Tall Theory of algebraic integers Dedekind Algebraic number theory FrÃ¶hlich, Taylor and Taylor Primes of the form x2 + ny2: Fermat, class field theory, and complex ... Cox A classical introduction to modern number theory Ireland and Rosen Number fields Marcus Number theory Borevich, Shafarevich and Greenleaf Local fields Cassels Lectures on number theory Dirichlet and Dedekind Algebraic Number Theory Neukirch

Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

Assessment

The assessment for this course will be via a single take-home paper in April with 2 weeks to complete and submit online. There will be 3 questions, and a pass will be awarded if your script is judged to be correct on at least half of the material set.

Magc 072 take home exam

 Files: Exam paper Released: Monday 13 April 2015 (894.7 days ago) Deadline: Sunday 26 April 2015 (880.7 days ago)

Files

Files marked L are intended to be displayed on the main screen during lectures.

Week(s) File Exercise 1.pdf Exercise 2.pdf Exercise 3.pdf handout week 10 lecture.pdf L handout week 2 lecture.pdf L handout week 3 lecture.pdf L handout week 4 lecture.pdf L handout week 5 lecture.pdf L handout week 6 lecture.pdf L handout week 7 lecture.pdf L handout week 8 lecture.pdf L handout week 9 lecture.pdf L Week 10 lecture.pdf L Week 3 Lecture.pdf Week 4 Lecture.pdf Week 5 Lecture.pdf Week 6 lecture.pdf L Week 7 lecture.pdf L Week 8 lecture.pdf L Week 9 lecture.pdf L 0 handout week 1 lecture.pdf L 0 Week 1 Lecture.pdf L 0 Week 2 Lecture.pdf L