Back to Introduction to Galois Theory

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116 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 RLee

•Dec 8, 2016

It is a rare online course of advanced pure mathematics. Overall it is very good. It is not recommended to take other courses in parallel with this course as it will consume lots of time on external notes and references. I would recommend National Research University to open more similar courses on abstract algebra, like Groups, Rings, Fields and Modules as a bridge program to this course, or an extension of this course to Lie Algebra or Representation Theory.

By Dmitry N

•Apr 2, 2016

That's a great course for those who want to broaden their horizons in mathematics. Learning process involves a huge amount of useful information from different areas, such as linear algebra, fields and rings theory, and many more.

But I'd like to share some thoughts with possible readers of this review.

1) A learner should have a really strong mathematical background. I would say that it's a graduate-level course, and I couldn't recognize that from course description. Otherwise,

2) One needs a HUGE amount of time to learn all the material thoroughly. Just to clarify, I'm changing my job now, so I fortunately have a lot of free time. And I spend almost ALL of one for this course! :) That's great, I'm ready to do that, but how many people have that kind of possibility? Amount of given information is enough for a separate specialization in Coursera terms, so maybe it would be good to split this course into several separate ones.. for sure, it fits into a standard HSE university course. But I can say that I do have some background in maths, but a number of years it was unused, so it takes time to recover a lot of things. Just my thoughts, but I see how discussion in forum is fading away, guess it means that a lot of people start giving up.

3) And finally I would suggest to add some small quizzes within lectures as it's done, for example, in 'Learning How To Learn' course. In my opinion, it can help learners to catch more from videos.

So, 3 weeks left to finish this course, so I'm starting a next quiz! Thanks to everyone and happy learning! Or teaching :)

By Chao G

•Aug 15, 2016

Overall, this course has very solid content. In fact, this is one of the few (probably the only one) advanced mathematics course on Coursera. More specifically, in order to take this course, you need to have good understanding of group, field and ring. In other words, you are assumed to have taken general undergraduate level abstract algebra courses before.

However, there are some aspects of the course that are worth improving. First, probably due to the nature of online course, I personally find learning Galois Theory online very challenging. The lectures themselves make sense. The practice problems and quiz, however, sometimes do not seem to be reinforcing the lectures. The materials on tensor product seem to be unrelated as well. Usually these are covered in a course on commutative algebra. In addition, there was almost none communication within the discussion forums.

Overall, I would definitely recommend this course if you have good background in general abstract algebra. After all, Galois Theory has long been a capstone type of course. If you have some background but not solid, it is definitely still doable.

By Wesley R S

•Dec 30, 2019

1) great course material, it's rather hard to find lectures online.

2) Calls a lot of results "trivial" and "easy" or very "simple".

3) The teacher does not seem enthusiastic nor is her English particularly good.

4) Lectures aren't very structured, the lecturer moves from definition to example to lemma to remark without clearly indicating the boundaries.

By Xinwei Y

•May 3, 2016

It's a good topic. However the quality of the lectures are pretty low. In my opinion people could learn the topic more efficiently by reading Stewart's book.

By Christopher L

•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.

By Michael F

•Jun 23, 2017

Difficult class, but well worth the effort!

By Enrico P

•Jun 21, 2017

The course is really well done, with good and clear videos and sets of weekly exercises. A bit concise and require a good knowledge of basic Algebra and some expansion with chapter books/notes during the lectures to fix concepts and see more examples. A set of slides could have been useful but there's a recap available from a student in the course that does a relatively good job for this. For me Week 4 and part of Week 5 (tensor products and related stuff) were really hard to follow without a previous knowledge of the matter, and I expanded it reading some suggested chapter books and internet notes (particularly useful those by Keith Conrad at U.Conn. on tensor product). I will move to five stars if some more time will be dedicated to explain tensor products and apps and make handouts.

By Esteban C d V

•Feb 1, 2017

Several comments, in general the course is well conducted,it is not easy to follow, but this is natural as it a very abstract subject. The exercise sets (peer-reviewed) where very instructive, but I think that the quizzes where a little too easy in comparison. I missed a little a more intuitive view, a broader view of the subject as well as a review of more advanced applications and developments, such as infinite Galois groups, etc.

By Wolfgang G

•Apr 27, 2018

First of all, it is great that some more advanced courses such as this one are offered on coursera. Unfortunately, a lot of the potential of online learning was not realised in this one. The lectures were handwritten on a tablet, there was no additional reading material, and as the text was difficult to read, it is often necessary to relisten to the video when you just want to look up some detail. Also, the problem sets were not very well coordinated with the lecture. The forums are almost deserted, there do not seem to be any moderators around.

By Krishnakant A

•Sep 15, 2017

this a great course.One might wonder considering the length of this course that the content is not much, but once started ,one week's content is more than enough to keep us busy for whole the week. as well as the references perfectly go hand in hand.

By Lara d M

•Jul 7, 2020

The couse is complete, everything you need to learn the introduction to Galois Theory is in it. I would recomend for all math students in all levels, as long as you truly have the time to study.

By Prasanna M

•Jul 31, 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.

By Shalini G A M

•Oct 30, 2020

You need to work hard to complete this course. Awesome exercises. I learned a lot from this course.

By Hou T C

•Feb 27, 2017

An intriguing course.

Uncommonly advanced material.

Rare content on Coursera.

By Deliang Z

•Oct 21, 2020

Hard Course, but if you spend time on it, you will definitely learn a lot.

By Roi Z

•May 15, 2016

Very interesting course for poeple with knowledge in algebra.

By Ivan M

•Jun 7, 2017

Very interesting course where I got a lot

By NAZIA J

•Jun 27, 2020

It was a great experience.

By Pranav R

•Nov 16, 2017

A very interesting course

By Sony D

•May 24, 2017

Work hard, learn more.

By Dr. A S M

•May 23, 2018

A wonderful course!

By Rodrigo A T M

•Apr 7, 2016

Great course!

By petya

•Aug 20, 2018

perfect

By Hangfeng G

•Mar 16, 2020

the content is rich, though a little advanced. I strongly recommend this course to others, because I personally learned a lot from it.

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