Introduction to Galois Theory
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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. The students shall learn to compute Galois groups and 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. A weekly test and two more serious exams in the middle and in the end of the course. For the final result, tests count 40%, first (shorter) exam 20%, final exam 40%. Do you have technical problems? Write to us: [email protected]
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Rating | 3.7★ based on 30 ratings |
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Length | 10 weeks |
Effort | 9 weeks of study, 4-8 hours/week |
Starts | Feb 14 (111 weeks ago) |
Cost | $49 |
From | Higher School of Economics, National Research University Higher School of Economics, HSE University via Coursera |
Instructor | Ekaterina Amerik |
Download Videos | On all desktop and mobile devices |
Language | English |
Subjects | Mathematics |
Tags | Math And Logic |
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What people are saying
interesting course
A very interesting course 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.
Not a very good or interesting course and does not use standard notation for the subject.
Very interesting course for poeple with knowledge in algebra.
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not seem
The forums are almost deserted, there do not seem to be any moderators around.
The practice problems and quiz, however, sometimes do not seem to be reinforcing the lectures.
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more advanced
First of all, it is great that some more advanced courses such as this one are offered on coursera.
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.
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abstract algebra
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.
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.
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Rating | 3.7★ based on 30 ratings |
---|---|
Length | 10 weeks |
Effort | 9 weeks of study, 4-8 hours/week |
Starts | Feb 14 (111 weeks ago) |
Cost | $49 |
From | Higher School of Economics, National Research University Higher School of Economics, HSE University via Coursera |
Instructor | Ekaterina Amerik |
Download Videos | On all desktop and mobile devices |
Language | English |
Subjects | Mathematics |
Tags | Math And Logic |
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