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I recently decided to introduce myself to the field of Modern Algebra - in particular, Galois theory - and I found it absolutely beautiful! Thus I would really like to study something in Galois theory, which leads me to ask - do people still develop Galois theory? What else is there to learn in the subject?

I am inspired by questions like these: What kind of work do modern day algebraists do? and What do modern-day analysts actually do? and would love to learn your opinions, stories, etc!

Thanks in advance!

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    $\begingroup$ Yes, it is a very active research area that can be approached via combinatorics, number theory, representation theory or algebraic geometry. Some classical problems like the inverse Galois problem over $\mathbb Q$ are still unresolved. $\endgroup$ Aug 31 '20 at 21:35
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    $\begingroup$ Yes, there is active research. There are still lots of open questions about the inverse Galois problem. I know one person who recently started a research in probabilistic Galois theory. (which deals with stuff like distribution of polynomials of degree $n$ in $\mathbb{Q}[x]$ with Galois group $S_n$ and more such questions). Don't really know the details though. $\endgroup$
    – Mark
    Aug 31 '20 at 21:38
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    $\begingroup$ you can look up the description of the proof of Fermat's Last Theorem and see that Galois theory was used in it, iirc. $\endgroup$
    – releseabe
    Sep 1 '20 at 11:59
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I'm going to give a short, very simplified overview of something that I'm somewhat familiar with, though there are many other open streams in research that I don't have the experience to comment on.

Let $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ denote the absolute Galois group, i.e. the group of all field automorphisms $\overline{\mathbb{Q}}\to\overline{\mathbb{Q}}$ fixing the rationals. Equivalently, $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is the inverse limit of Galois groups $\text{Gal}(L/\mathbb{Q})$ of finite Galois extensions $L/\mathbb{Q}$, so in a certain sense it is made up of all finite Galois groups over $\mathbb Q$.

Perhaps the most well-known open problem in Galois theory is

What is the structure of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$?

An approach to this problem is through the famous Langlands program. A different approach was outlined by Grothendieck in his, also relatively well known, Esquisse d'un programme.

There Grothendieck notes that $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ has a faithful action on the collection of graphs embedded on compact surfaces, which he calls dessins d'enfants (children's drawings) due to their apparent simplicity. If one can understand this action, then in principle one can represent the elements of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ as permutations of dessins d'enfants. Thus one of the main open problems of the theory of dessins d'enfants is to

Classify enough invariants of dessins d'enfants so that any two orbits of the action of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ can be distinguished.

Shortly after Grothendieck's Esquisse went into circulation, Drinfeld proved that $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ injects into the so-called Grothendieck-Teichmuller group, which has an explicit description in terms of generators and relations. Hence one other open problem is

Is $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ isomorphic to the Grothendieck-Teichmuller group?

These are very difficult problems which raise further questions still unresolved, for example how can one compute a dessin d'enfant efficiently. Furthermore, theoretical physicists are also interested in Galois theory: Drinfeld's introduction of the Grothendieck-Teichmuller group was motivated by mathematical physics, and dessin d'enfants have already appeared in physics under a different name, dimer models.

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    $\begingroup$ Thanks for this answer. The site definitely needs more users who have seen this stuff. $\endgroup$ Sep 27 '20 at 14:32
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There are several generalizations of Galois Theory in progress. Some of them are of groups acting on commutative and non-commutative rings (other than fields), groups acting partially on rings, groupoids acting globally and partially on rings, and much more. We have a group of students and professors who work with that type of variants in my university (Federal University of Rio Grande do Sul - UFRGS). Some beautiful papers that you can look to know more about it (they are a little difficult, but are just examples of what you can do):

Partial groupoid actions: globalization, Morita theory and Galois theory

The Galois correspondence theorem for groupoid actions

Partial actions and Galois theory

A Characterisation for a groupoid Galois extension using partial isomorphisms

A Galois-Grothendieck-type correspondence for groupoid actions

Galois correspondences for partial Galois Azumaya extensions

Hope you enjoy it!

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