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.

I’ve been watching MOOCs for several years and this is one of the few I’ve come across that covers some more advanced mathematical topics. I’m curious to see how it turns out and what type of interest/results it returns.

It’s being offered by National Research University – Higher School of Economics (HSE) in Russia.

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## Author: Chris Aldrich

I'm a biomedical and electrical engineer with interests in information theory, complexity, evolution, genetics, signal processing, theoretical mathematics, and big history.
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@ChrisAldrich also the mathematical underpinnings of a lot of encoding/decoding used in digital communications

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@wa7iut Naturally!

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Introduction to Galois Theory | Coursera boffosocko.com/2016/09/11/int…

Reminder: Introduction to Galois Theory started yesterday | Coursera http://boffosocko.com/2016/09/11/introduction-to-galois-theory-coursera

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Reminder: Introduction to Galois Theory started yesterday | Coursera http://boffosocko.com/2016/09/11/introduction-to-galois-theory-coursera

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I’d tried that course a while back, I needed more abstract algebra knowledge sadly.

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You don’t really need all of Abstract Algebra, just a reasonable grounding in Group theory.

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You could start with Benedict Gross’ class on Abstract Algebra instead: extension.harvard.edu/open-learning-…

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when I said abstract algebra I meant group theory. I’ll have a break soon so I might just do that.

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