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I'm trying to ask this question in the same vein as the various "what is the point of X field" that appear often on this site.

I've just finished a first course in Galois theory. Results that are often proved in a first course (indeed, including mine) include ancient problems such as squaring the circle, doubling the cube, trisecting the angle are impossible, and the insolvability of the quintic.

But other than that, what applications are there to Galois theory? What other areas of mathematics tend to rely heavily on the technology developed in Galois theory?

Robin
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  • Number theory relies "heavily" on it. Galois theory is an important tool for studying the arithmetic of number fields and function fields. Further keywords are the absolute Galois group ${\rm Gal}(\overline{\Bbb Q}/\Bbb Q)$ , Galois theory of Riemann surfaces, Galois theory and algebraic geometry and so on. This site has some posts on it, e.g., this post. – Dietrich Burde Dec 08 '23 at 22:02
  • Parts of Galois theory appeared in pretty much any advanced algebra course I took. It is also a very active area of research. – Mark Dec 08 '23 at 22:07
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    A somewhat related thread. There the focus is on cryptography, but in a comment I described a telecommunication problem that can be thought of as an application of Galois theory, in a somewhat vague sense :-) – Jyrki Lahtonen Dec 11 '23 at 08:51

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