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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

Autumn 2011 (Monday, October 10 to Friday, December 16)

Timetable
  • Fri 12:05 - 12:55

Lecturer


Neil Dummigan
Email n.p.dummigan@sheffield.ac.uk
Phone (0114) 2223713
vcard
Photo of Neil Dummigan


Students


Photo of Stacey Aston
Stacey Aston
(Newcastle)
Photo of Tom Brookfield
Tom Brookfield
(Birmingham)
Photo of Anthony Chiu
Anthony Chiu
(Manchester)
Photo of Daniel Fretwell
Daniel Fretwell
(Sheffield)
Photo of Tom Harris
Tom Harris
(Southampton)
Photo of Andrew Jones
Andrew Jones
(Sheffield)
Photo of Matthias Krebs
Matthias Krebs
(East Anglia)
Photo of Nil Mansuroglu
Nil Mansuroglu
(Manchester)
Photo of Keith McCabe
Keith McCabe
(Durham)
Photo of Gregory McKay
Gregory McKay
(East Anglia)
Photo of Laura Phillips
Laura Phillips
(Manchester)
Photo of Alan READING
Alan READING
(Birmingham)
Photo of Amin Saied
Amin Saied
(Southampton)
Photo of Konstantinos Tsaltas
Konstantinos Tsaltas
(Sheffield)
Photo of David Ward
David Ward
(Manchester)
Photo of Yue  Wu
Yue Wu
(Loughborough)
Photo of Stefanie Zegowitz
Stefanie Zegowitz
(East Anglia)


Prerequisites


Elementary number theory (primes, linear congruences). Rings and groups, including irreducibles, units, Euclidean domains, quotient groups, 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.

Bibliography


Algebraic number theory and Fermat's last theorem Stewart and Tall
Theory of algebraic integers Dedekind
Algebraic number theory Fröhlich, 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
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Assessment


There will be a single assignment, with one big question, which has to be passed in order to pass the module. It will be set on 9th December 2011, and due in at noon on 10th January 2012.

Assignments


Cyclotomic Fields

Files:Exam paper
Deadline: Tuesday 10 January 2012 (933.4 days ago)