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I was able to reduce the problem to solving the quintic equation $$x^5+x^4-4x^3-3x^2+3x+1=0,$$ which seems to be the minimal polynomial of $2\cos(2\pi/11)=e^{2\pi i/11}+e^{-2\pi i/11}$. But I couldn't find a method to go further with my calculation. I haven't studied Galois theory, although I know some basic field theory.

Finn Bolton
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  • I'm not quite certain want you want. Galois theory (with the addition of results by Kummer) does say that if $K=\Bbb{Q}(\sqrt{-11},e^{2\pi i/5})$, then there exists an element $z\in K$ such that $e^{2\pi i/11}\in K(\root5\of z)$, enabling us to write $e^{2\pi i/11}$ in terms of $\sqrt{-11}$, $e^{2\pi i/5}$ and $\root5\of z$, which is probably what you are looking for. But it would take a bit of my time to figure out $z$ and the sought after expression. – Jyrki Lahtonen Feb 27 '24 at 07:06
  • Your experience on the site will probably improve, if you take a look at our abridged guide for new askers. You may have noticed that some reviewers already voted to put your question on hold for it is not quite up to the expected standards. I'm sure you can do a better job, and in that link others have collected pointers about what's expected. – Jyrki Lahtonen Feb 27 '24 at 07:14
  • Ok. Put such pieces of information into the question body (like I just did). That way you make a more convincing case. It's just one of the "charms" of how this site works :-). Thinking a bit more about the actual question. Kummer's method also tells how to get that cosine in terms of a fifth root of some element of the cyclotomic field $\Bbb{Q}(e^{2\pi i/5})$. So we can possibly leave out $\sqrt{-11}$. But I gotta work now. We have many users capable of helping you. If not, I may come back to this later. – Jyrki Lahtonen Feb 27 '24 at 07:26
  • What is the source of the problem? – D S Feb 27 '24 at 08:08
  • @DS I was able to calculate the seventh roots of unity by reducing the problem to solving the cubic equation $x^3 + x^2 - 2x - 1 = 0$, however I couldn't do the same trick with the eleventh roots as I don't know how to solve quintic equations. – Finn Bolton Feb 27 '24 at 08:09
  • Since there is no general method for quintics – D S Feb 27 '24 at 08:11
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    I think I have seen a similar question, but I'm neither sure nor can I find it. – D S Feb 27 '24 at 08:15
  • @DS My recollection is similar. I thought I had answered a similar question about $e^{2\pi i/7}$, but couldn't find it. May be I only answered in a comment as I couldn't be arsed to do the calculation to the end :-) – Jyrki Lahtonen Feb 27 '24 at 08:56
  • An earlier thread on the seventh root of unity. – Jyrki Lahtonen Feb 27 '24 at 09:36
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    Earlier stuff about the eleventh root – Jyrki Lahtonen Feb 27 '24 at 09:40
  • Abel first and Galois after etc, etc..... Maybe your particular quintic admits the formula you desire but why? Meanwhile follow reading Quora. – Ataulfo Feb 28 '24 at 12:27
  • Does this answer your question? Radical representation of $\cos\frac {2\pi}{11}$ – Benjamin Wang Feb 28 '24 at 13:47
  • @BenjaminWang Not much, I don't understand the method of solution. – Finn Bolton Feb 29 '24 at 17:28

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