Master Training Program
Royal Academy of Cambodia / CIMPA
October 2006

Coopération Mathématique Interuniversitaire Cambodge France,
PISF (Programme International de Sciences Fondamentales)
de l'UNESCO (United Nations Educational, Scientific and Cultural Organisation)

Introduction to algebraic number theory
Phnom Penh, October 2-27, 2006 (60 hours)

This course (60 hours) is a relatively elementary course which requires minimal prerequisites from Commutative Algebra for its understanding.

Syllabus:

0. Introduction: Diophantine Equations
Linear equations, systems of linear equations.
Pythagoreas quadratic equation. Pell equation. Continued fraction expansion.
The field of Gaussian numbers. Arithmetic in the ring of Gaussian integers. Euclidean rings, principal rings, UFD (unique factorization domains).

1. Field extensions
1.1 Algebraic and transcendental extensions. Simple extension defined by an irreducible polynomial. Splitting field of a polynomial. Algebraic closure.
1.2 Field homomorphisms. Conjugates of an algebraic number. Separable extensions.
1.3 Galois extensions, Galois theory.

2. Integrality
2.1 Integrality over rings. Integral closure, integrally closed rings. Ring of integers of a number field.
2.2 Characteristic polynomial, norms and traces
2.3 Structure theorem of a finitely generated module over a principal ring. Integral basis. Discriminant.
2.4 Cyclotomic fields
2.5 Units of rings of algebraic integers

Bibliographic references:

The main reference is a lecture notes by Ivan B. Fesenko
ps file (432K)
pdf file (193K)

Further references are available on the Online number theory lecture notes website, Selected from the Descriptions of areas/courses in number theory, lecture notes

Among other relevant references from the same source are

MAS4002: Algebraic Number Theory, Course notes by Robin Chapman, University of Exeter

Lecture notes on algebraic number theory by Jerome Hoffman

MP473 Number Theory IIIH/IVH, Semester 2, 2000 by Keith Matthews

Algebraic Number Theory and commutative algebra, lecture notes by Robert Ash

Math 254B (Number Theory), lecture notes on class field theory, abelian extensions of number fields etc by Kiran Kedlaya

Algebraic number theory course book by William Stein

Algebraic number theory course notes by Tom Weston

Many books are relevant to this topic, including in english
Jody Esmonde and M. Ram Murty Problems in Algebraic Number Theory (Springer Verlag)
David S.Dummit and Richard M. Foote Abstract algebra (Prentice Hall)
et en français
Pierre Samuel Théorie algébrique des nombres (Hermann)
Daniel Duverney Théorie des Nombres, Cours et exercices corrigés (Dunod).



Exam / Examen
Examen écrit du 26 octobre 2006, sujet (en français) 90 Ko
Examen écrit du 26 octobre 2006, corrigé (en français) 112 Ko

Written exam, October 26, 2006 (in english) 87 Ko
Written exam, October 26, 2006, solution (in english) 111 Ko