Back to Introduction to Galois Theory

4.3

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113 ratings

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35 reviews

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.
We first shall survey the basic notions and properties of field extensions: algebraic, transcendental, finite field extensions, degree of an extension, algebraic closure, decomposition field of a polynomial.
Then we shall do a bit of commutative algebra (finite algebras over a field, base change via tensor product) and apply this to study the notion of separability in some detail.
After that we shall discuss Galois extensions and Galois correspondence and give many examples (cyclotomic extensions, finite fields, Kummer extensions, Artin-Schreier extensions, etc.).
We shall address the question of solvability of equations by radicals (Abel theorem). We shall also try to explain the relation to representations and to topological coverings.
Finally, we shall briefly discuss extensions of rings (integral elemets, norms, traces, etc.) and explain how to use the reduction modulo primes to compute Galois groups.
PREREQUISITES
A first course in general algebra — groups, rings, fields, modules, ideals. Some knowledge of commutative algebra (prime and maximal ideals — first few pages of any book in commutative algebra) is welcome. For exercises we also shall need some elementary facts about groups and their actions on sets, groups of permutations and, marginally,
the statement of Sylow's theorems.
ASSESSMENTS
A weekly test and two more serious exams in the middle and in the end of the course. For the final result, tests count approximately 30%, first (shorter) exam 30%, final exam 40%.
There will be two non-graded exercise lists (in replacement of the non-existent exercise classes...)
Do you have technical problems? Write to us: coursera@hse.ru...

PM

Jul 30, 2020

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.

CL

Jun 15, 2016

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

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By Troy W

•Mar 12, 2018

The teacher is good at explaining things.

It is best you take an algebra course for prerequisite.

By TH

•Mar 23, 2016

Actually rigorous and non-trivial maths.

By Vineet G

•May 30, 2016

The material is very interesting, but the course goes very fast, and the presentation is dry.

By 李宗桓

•Apr 24, 2016

我对此评分不高的主要原因是因为授课太难，另外中间tensor product of modules很难又很偏。

By Musa J

•Jan 28, 2018

Please show visual examples, diagrams to start with; -Class notes should be ready before class starts; first motivational examples then definitions please. https://www.youtube.com/watch?v=8qkfW35AqrQ Visual Group Theory, Lecture 6.6: The fundamental theorem of Galois theory. For now pls unenroll me from this course.

By Corey Z

•Jan 25, 2018

Very Hard to follow. She is constantly writing things while teaching. She could have written down everthing before class.

By Ryan B

•Sep 11, 2017

Not a very good or interesting course and does not use standard notation for the subject.

By Rod B

•Apr 18, 2016

Hopeless ! Less clear and understandable than simply reading a textbook.

A waste of time.

By Maneesh N

•Oct 29, 2019

Instructor is monotonous.

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