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What are some interesting nonstandard applications of Galois Theory?

For instance, I would call these applications standard: impossibility of solving quintic, squaring circle, doubling cube.

While Bruno Joyal's proof in A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational , I would consider nonstandard.

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Spencer Hyman
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  • Depends on what "Galois theory" means. Would you consider study of Galois representations (representation of $\text{Gal}(\overline{\Bbb Q}/\Bbb Q)$ in $\text{GL}_n(\Bbb F_p)$) "Galois theory"? Then these representations have a lot of applications in number theory, e.g., Fermat's last theorem. – Balarka Sen Jan 12 '15 at 08:15
  • Let's restrict the question to applications of basic Galois theory: what one would learn in a first algebra course. – Spencer Hyman Jan 12 '15 at 08:22
  • There is an application of basic Galois theory which says that if a quintic over $\Bbb Q[x]$ can be solved in terms of $\mathcal{EL}$-numbers, then Schanuel's conjecture is false. See this paper by T. Y. Chow. Would you consider this nonstandard? – Balarka Sen Jan 12 '15 at 08:31
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    Impossibility of squaring the circle or doubling the cube are not applications of Galois theory. The first one is based on $\pi$ being transcendental and the second is based on constructible numbers having degree a power of $2$ over $\mathbf Q$, and neither of these use Galois theory. Are you confusing Galois theory with field theory? – KCd Jan 12 '15 at 09:48
  • It would help if you explained what your own background is in Galois theory (what applications have you seen?). If you go further into number theory then you see Galois theory used all over the place. Is all of that "standard"? I am not sure, as the example you consider standard is about rationality and powers. Does "nonstandard" mean "not easily found in certain kinds of books?" – KCd Jan 12 '15 at 09:51
  • Hi KCd, yes, I am just looking for applications that are not easily found in books. Interesting number theory applications would be fair game. – Spencer Hyman Jan 12 '15 at 14:36
  • @Dongerz "Interesting number theory applications" comes up only if you allow things like absolute Galois groups to act on something. – Balarka Sen Jan 17 '15 at 11:14

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