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I'm enjoying reading about Janelidze's categorical Galois theory, which gives as a special case the usual theorems of Galois descent (along torsors). The approach I took was just with covering space theory in mind, using locally connected categories for intuition for the very general type of adjunctions there.

I think I feel reasonably comfortable with the categorical Galois theorem, and I've been asked to give a very short presentation about it. I wanted to include applications of Galois descent to show people that it's not empty abstraction. (A few of the few people attending are students who like me don't know any non-trivial algebraic geometry.)

What are some neat applications of Galois descent which are fairly simple to explain or describe (preferably geometrically)?

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    $\begingroup$ Out of curiosity, why did you teach yourself categorical Galois theory without first being aware of some of the interesting applications of traditional Galois descent? In what sense does one become comfortable with a theorem without knowing how to illustrate that it is not empty abstraction? $\endgroup$
    – nfdc23
    Commented Jan 22, 2017 at 1:16
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    $\begingroup$ I like categories and thought it's cool a single formalism encompasses different fields. I never felt I needed explicit classical Galois stuff for motivation. I only mean I'm comfortable with the categorical stuff in the sense I feel I understand the covering space aspect of the abstractions. The fact esteemed category theorists wrote a book on the topic was enough to make me feel it's not empty abstraction :) $\endgroup$
    – Arrow
    Commented Jan 22, 2017 at 1:30
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    $\begingroup$ I give some simple examples here: qchu.wordpress.com/2015/11/17/forms-and-galois-cohomology $\endgroup$
    – Qiaochu Yuan
    Commented Jan 22, 2017 at 2:41
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    $\begingroup$ OK, though I recommend having more tangible prior motivation than "a famous person wrote about this" before deciding to dive into a topic. Anyway, section 1 of Chapter III of Serre's classic book Galois cohomology (building on the general formalism he sets up in section 5 of Chapter I) is the standard succinct reference on important basic examples of Galois descent. The more recent book Central Simple Algebras and Galois Cohomology by Gille and Szamuely contains a wealth of significant examples and applications worked out in extensive detail (see its table of contents). $\endgroup$
    – nfdc23
    Commented Jan 22, 2017 at 3:13
  • $\begingroup$ @QiaochuYuan In a recent answer you also described a quite accessible example of how Galois descent can be used to obtain interesting information about Hopf algebras over non-algebraically-closed fields. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 22, 2017 at 8:11

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