Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Hermite-Minkowski Theorems: Number Fields and Ideal Classes

In courses (2)

MATH-482: Number theory I.a - Algebraic number theory

Algebraic number theory is the study of the properties of solutions of polynomial equations with integral coefficients; Starting with concrete problems, we then introduce more general notions like algebraic number fields, algebraic integers, units, ideal c ...

MATH-317: Algebra V - Galois theory

Galois theory lies at the interface of Field Theory and Group Theory. It aims to describe the algebraic symmetries of fields. We will focus on Galois theory for finite field extensions and some applications. ...

Description

This lecture covers Hermite-Minkowski theorems related to number fields, discussing the ramification of number fields, the finiteness of certain number fields, and the product of intervals and circles. It also explores the concept of ideal classes, the abelian group of units, and the group of roots of unity contained in a number field.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Instructors (3)

Ph. Michel's main research interest lie in the field of analytic number theory and range over a variety of techniques and methods which include: arithmetic geometry, exponential sums, sieve methods, automorphic forms and allied representations, L-functions and more recently ergodic theory. Ph. Michel is a former student of ENS Cachan and obtained his PhD in Universté Paris XI in 1995 under the guidance of E. Fouvry. From 1995 to 1998 he was maître de conférence at Universté Paris XI and full professor at Université Montpellier II until 2008 then when he joined EPFL. Ph. Michel was awarded the Peccot-Vimont prize, has been member of the Institut Universitaire de France and wa invited speaker at the 2006 International Congress of Mathematician.

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Related concepts (32)

Ideal class group

In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK/PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of K. The order of the group, which is finite, is called the class number of K. The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group.

Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

Fractional ideal

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity. Let be an integral domain, and let be its field of fractions.

Finitely generated abelian group

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a generating set of or that generate . Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. The integers, , are a finitely generated abelian group. The integers modulo , , are a finite (hence finitely generated) abelian group.

Ramification (mathematics)

In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. Branch point In complex analysis, the basic model can be taken as the z → zn mapping in the complex plane, near z = 0.

Related lectures (33)

Ideal Class Group Relations

Covers the relations between the ideal class group and proper fractional ideals.

The Discriminant and Ideal Class Group in Mathematics

Explores the discriminant in matrices, ideal class groups, and optimal embeddings in mathematics.

Linear Molecules: Carbon

Covers the properties of linear molecules in carbon and discusses neutral elements, associative law, and conjugate elements.

Functor Hom: Abelian Groups

Explores the Hom functor in Abelian groups, focusing on its construction and properties.

Exact Sequences: Torsion and Divisibility