A very beautiful classical theory on field extensions of a certain type (Galois extensions) initiated by Galois in the 19th century. Explains, in particular, why it is not possible to solve an equation of degree 5 or more in the same way as we solve quadratic or cubic equations. You will learn to compute Galois groups and (before that) study the properties of various field extensions.

## About this Course

## Offered by

### HSE University

HSE University is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communicamathematics, engineering, and more.

## Syllabus - What you will learn from this course

**1 hour to complete**

## Introduction

This is just a two-minutes advertisement and a short reference list.

**1 hour to complete**

**2 hours to complete**

## Week 1

We introduce the basic notions such as a field extension, algebraic element, minimal polynomial, finite extension, and study their very basic properties such as the multiplicativity of degree in towers.

**2 hours to complete**

**2 hours to complete**

## Week 2

We introduce the notion of a stem field and a splitting field (of a polynomial). Using Zorn's lemma, we construct the algebraic closure of a field and deduce its unicity (up to an isomorphism) from the theorem on extension of homomorphisms.

**2 hours to complete**

**4 hours to complete**

## Week 3

We recall the construction and basic properties of finite fields. We prove that the multiplicative group of a finite field is cyclic, and that the automorphism group of a finite field is cyclic generated by the Frobenius map. We introduce the notions of separable (resp. purely inseparable) elements, extensions, degree. We briefly discuss perfect fields. This week, the first ungraded assignment (in order to practice the subject a little bit) is given.

**4 hours to complete**

**2 hours to complete**

## Week 4

This is a digression on commutative algebra. We introduce and study the notion of tensor product of modules over a ring. We prove a structure theorem for finite algebras over a field (a version of the well-known "Chinese remainder theorem").

**2 hours to complete**

## Reviews

### TOP REVIEWS FROM INTRODUCTION TO GALOIS THEORY

A difficult course for me, personally, but that makes it all the more worth it! Taking this course has helped me learn more I thought I would. Definitely recommended.

Thank You Prof Ekaterina Amerik for this course. I understand with your explaination. Maybe i must study again and again for understand this material.

Outstanding course so far - a great refresher for me on Galois theory. It's nice to see more advanced mathematics classes on Coursera.

The teacher is good at explaining things. It is best you take an algebra course for prerequisite.

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