返回到 Introduction to Galois Theory

4.3

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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...)...

創建者 CL

•Jun 16, 2016

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

創建者 RZ

•May 15, 2016

Very interesting course for poeple with knowledge in algebra.

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26 個審閱

創建者 petya

•Aug 20, 2018

perfect

創建者 Dr. A. S. Muktibodh

•May 24, 2018

A wonderful course!

創建者 Wolfgang Globke

•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.

創建者 Troy Woo

•Mar 12, 2018

The teacher is good at explaining things.

It is best you take an algebra course for prerequisite.

創建者 Musa Jahanghir

•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.

創建者 Corey Zhang

•Jan 25, 2018

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

創建者 Pranav Rao

•Nov 16, 2017

A very interesting course

創建者 Krishnakant Ammanamanchi

•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.

創建者 Ryan Birmingham

•Sep 11, 2017

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

創建者 Michael J Fisher

•Jun 23, 2017

Difficult class, but well worth the effort!