# ALGEBRAIC FIELD EXTENSIONS

## Learning outcomes of the course unit

The student should acquire a good knowledge about rings of integers in number fields and about the basic notions of algebraic number theory. The student should also be able to apply such knowledge to the investigation of various algebraic extensions (in particular quadratic and cyclotomic) to solve factorization problems and understand some aspects of more complicated problems like Fermat's Last Theorem.

After the lectures the student should be able to present the topics of the course with clarity and precision and with an appropriate specific scientific language and to improve his/her knowledge in algebraic number theory by consulting the existing literature on the subject.

Good knowledge of the basic aspects of algebraic number theory.

## Prerequisites

Knowledge of the basic algebraic strucctures (groups, rings and fields).

## Course contents summary

The course describes general properties of Dedekind domains and then focuses on the particular case of the rings of integers of number fields (finite extensions of the rationals). We present the basic tools of algebraic number theory and prove theorems on the structure of rings of integers and on the factorization of primes which allow the student to have a good understanding of some aspects of classical problems like Fermat's Last Theorem.

Integral extensions: algebraic elements, minimal polynomials, primes in integral extensions, "going up" and "going down" theorems, integrally closed domains.

Dedekind domains: noetherian rings, local Dedekind domains, unique factorization of ideals, class group.

Number fields: finite extensions of the rationals, embeddings in the complex numbers, norm and trace maps, discriminant, ring of integers, examples: quadratic, cubic and cyclotomic fields.

Factorization of primes: factorization in rings of integers, ramification index and inertia degree, Kummer's theorem, Dedekind's theorem, factorization and Galois theory, examples: quadratic and cyclotomic fields.

## Course contents

Integral extensions: algebraic elements, minimal polynomials, primes in integral extensions, "going up" and "going down" theorems, integrally closed domains.

Dedekind domains: noetherian rings, local Dedekind domains, unique factorization of ideals, class group.

Number fields: finite extensions of the rationals, embeddings in the complex numbers, norm and trace maps, discriminant, ring of integers. Examples: quadratic, biquadratic, cubic and cyclotomic fields.

Factorization of primes: factorization in rings of integers, ramification index and inertia degree, Kummer's theorem, Dedekind's theorem, factorization and Galois theory. Examples: quadratic and cyclotomic fields.

During the lectures we shall provide the basic notions of commutative algebra and Galois theory required for the understanding of the main topics of the course.

## Recommended readings

D.A. Marcus "Number Fields" Universitext, Springer-Verlag.

J.S. Milne "Algebraic Number Theory" http://www.jmilne.org/math/CourseNotes/ant.html

J. Neukirch "Algebraic Number Theory" Gr. der math. Wissenschaften 322, Springer-Verlag.

M.R. Murty - J. Esmonde "Problems in Algebraic Number Theory" GTM 190, Springer-Verlag.

D.A. Marcus "Number Fields" Universitext, Springer-Verlag.

M.R. Murty - J. Esmonde "Problems in Algebraic Number Theory" GTM 190, Springer-Verlag.

J.S. Milne "Algebraic Number Theory" http://www.jmilne.org/math/CourseNotes/ant.html

## Teaching methods

The preferred teaching tool for the knowledge development are the 4 weekly hours of lectures: during those hours we present the theory and a vast library of examples and exercises/applications.

Taking notes is seen as part of the learning process.

Lectures

## Assessment methods and criteria

The assessment of learning is done in classic way, through the evaluation of an oral interview on all the topics treated during the lectures. In the colloquium, the student must be able to independently conduct demonstrations and solve exercises using an appropriate algebraic language and a proper mathematical formalism.

The outcome is positive if the student obtains a grade of (at least) 18. The maximal grade is 33 and a student who obtains more than 30 points is awarded a 30 cum laude grade.

Oral exam