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:

• Describe the fundamental concepts of field theory;

• Apply the main constructions, also in new contexts;

• Prove propositions and theorems in field theory;

• Communicate the course content with rigor and proper language;

• Recognize in concrete examples the properties of groups and fields studied;

• Compute in some concrete examples splitting fields, Galois groups, and other mathematical objects covered in the course;

• Apply the Galois correspondence to solve problems in field theory.

PREREQUISITES

Complete mastery of the contents of algebra courses from the first two years is required.The definition of field. Complex numbers. Basic notions of algebra, including: Gauß's lemma; Eisenstein criterion; group actions.

TEACHING METHODS

Lectures and frontal exercises. Video lessons on Teams only in rare cases of necessity.
Students with certifications of learning disabilities (DSA), disabilities, or other special educational needs are advised to contact the instructor at the beginning of the course to agree on teaching methods that, respecting the course objectives, take into account individual learning styles 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

LESSONS

LESSONS START

According to the schedule of the Corso di Studi

https://corsi.unige.it/corsi/9011/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written test and an oral test.
The written test will last between 2 and 3 hours and will involve writing proofs.
Passing the written test allows admission to the oral exam within the same exam session (summer/autumn/winter).

ASSESSMENT METHODS

The written test will consist of proof-based problems to verify the ability to calculate concrete mathematical objects seen in class and to apply theoretical notions to problem-solving.
The oral exam will cover all the lesson content, including definitions and proofs seen during lectures, and possibly new exercises; clarity of exposition and accurate use of mathematical terminology will be taken into account.

FURTHER INFORMATION

More/more current information on Aulaweb. For anything else contact the professor.