OVERVIEW

This course presents the foundations of field theory and Galois theory, a mathematical theory developed at the beginning of the nineteenth century to study the solvability of algebraic equations.

Motivated by classic problems such as the search for a formula for the solutions of fifth degree equations or the construction with ruler and compass of a regular polygon with 7 sides, we will show how to attach a group of permutations to a field extension. This establishes a very deep correspondence between groups and fields, which provides a dictionary for transporting concepts and properties from field theory to group theory and vice versa.

In this way we will be able to translate field theory questionss such as the problem of finding solutios of fifth degree equations into a group theory problem such as the existence of certain subgroups of S_5, and settle it in the world of groups.

AIMS AND CONTENT

LEARNING OUTCOMES

o provide an in-depth knowledge of field extensions and of Galois theory, in particular to deepen some applications of cyclotomic fields and radical solvability of algebraic equations.

AIMS AND LEARNING OUTCOMES

The purpose of the course is to:

• explain the basic properties of field extensions;
• define the notion of Galois group;
• show the correspondence between properties of field extensions and properties of the Galois groups attached to them;
•  apply this correspondence to solve (in the negative) some classical problems of algebra and Euclidean geometry.

At the end of the course, the student will be able to:

• repeat the definitions learned;
• recognize in concrete examples the properties of groups and fields studied;
• calculate in some concrete examples splitting fields, Galois groups and other mathematical objects seen in theory;
• apply the Galois correspondence to solve problems of field theory.

PREREQUISITES

The definition of field. Complex numbers. Basic notions of algebra, including: Gauß's lemma; Eisenstein criterion; group actions.

TEACHING METHODS

Traditional teaching, if the pandemic allows.

Students with SLD, disability or other special educational needs certification are advised to contact the teacher at the beginning of the course to agree on teaching and exam methods that, in compliance with the teaching objectives, take into account the modalities learning opportunities and provide suitable compensatory tools.

SYLLABUS/CONTENT

Field extensions and their basic properties.

Algebraic closure of a field: existence and uniqueness. Kronecker construction.

Normal splitting fields and extensions.

Separable, inseparable and purely inseparable extensions. Primitive element theorem.

Galois Extensions. Galois group and Galois correspondence for finite extensions.

Profinite groups and Krull topology. Galois correspondence for infinite extensions.

Galois group of an equation. Cyclotomic extensions. Generic equation of degree n.

Linear independence of characters. Trace and norm. Hilbert's Theorem 90. Hints of group cohomology. Cyclical extensions and Kummer theory.

Solvable groups. Solvable extension andradical extensions.

Further examples and applications.

RECOMMENDED READING/BIBLIOGRAPHY

Algebra S. Bosch

Algebra S. Lang

Algebra M. Artin

Class Field Theory J. Neukirch

TEACHERS AND EXAM BOARD

Exam Board

STEFANO VIGNI (President)

FRANCESCO VENEZIANO

MATTEO PENEGINI (President Substitute)

LESSONS

LESSONS START

According to the schedule of the Corso di Studi

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

If the pandemic situation allows it, the exam will consist of a written and an oral part. Passing the written test will allow admission to the oral test within the same exam session.

ASSESSMENT METHODS

The written test will consist of problems to assess the ability to

• compute the mathematical objects seen in class in concrete cases 
• apply theoretical notions to problem solving

The oral exam will focus on the content of the lectures and will also take into account the clarity of presentation and the accurate use of scientific terminology.

Exam schedule

FURTHER INFORMATION

More/more current informations on Aulabew