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I have edited my question since I clearly wasn't communicating well what I was looking for.

I am given $\theta$ such that $e^{i\theta}$ is a root of unity. We require $$\theta=q\pi,\quad q\in\mathbb{Q}.$$ Clearly, $\theta$ can't be rational.

Given irrational $\theta$, how does one check if $\theta/\pi$ is rational/irrational?

So, I want to know what are methodology one takes to figure out if $\theta/\pi$ is rational. This post appears to give an example of this, but I don't know what rings and Gaussian integers are. Perhaps the method can be explained in simpler terms?

Edit: @TedShifrin asks: "how precisely are you given $\theta$?". I am considering a function that has a parameter $q$. For $q$ that are roots of unity, this function isn't well defined. I was hoping there was a general method of determining if a particular parameter $q$ is a root of unity. I can accept that general methods don't exist. If this is the case, I would of course settle for less general ones, like those used in the link above; I simply don't understand what the method they used was. But I don't have a particular $\theta$ to give you; I would like to learn which classes of $\theta$ can we determine one way or the other.

David Raveh
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  • $e$ do not lies $\mathbb Q$ and $\pi\mathbb Q$. – MH.Lee Aug 03 '23 at 03:35
  • Hint: $(e^{i\theta})^n=e^{in\theta}$ and $e^{ix}=1$ implies $x$ is a multiple of $2\pi$ (use $e^{ix}=\cos x+i\sin x$). – coiso Aug 03 '23 at 03:40
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    So how precisely are you "given" $\theta$? – Ted Shifrin Aug 03 '23 at 04:15
  • I feel like I'm misunderstanding your question, because it seems obvious to me that if $\theta = q\pi$, then $\theta/\pi = q$; in other words, since $\theta$ is by definition a rational multiple of $\pi$, then of course $\theta/\pi$ is equal to said rational number. What am I missing? – H. sapiens rex Aug 03 '23 at 04:18
  • @H.sapiensrex I want to know what method is available to me to check, given an arbitrary irrational theta, whether theta/pi is rational – David Raveh Aug 03 '23 at 04:22
  • @DavidRaveh I see. So what does $\theta$ being given as $q\pi$ have to do with the question? – H. sapiens rex Aug 03 '23 at 04:25
  • @TedShifrin I have edited my question to explain this. I don't have a specific example; I am looking for classes for which methods exist to determine one way or the other. – David Raveh Aug 03 '23 at 04:52
  • There is no answer to your question other than to say that $e^{i\theta}$ is a root of unity if and only if $\theta/\pi\in\Bbb Q$. You still have not given any clue as to how you're handed an irrational $\theta$. All the fancier number theory and algebra to which you refer applies when you're given $a+bi\in\Bbb C$, not when you're given $e^{i\theta}$. – Ted Shifrin Aug 03 '23 at 05:08
  • @TedShifrin I didn't understand what you meant when you wrote "how are you given theta". I am ok with being given $a+bi$, so that $\theta=\arctan(b/a)$...I don't see how this helped. – David Raveh Aug 03 '23 at 05:14
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    Yeah, we're going around and around in circles. There is no easy answer. Basically you have to decide when the number $z=a+bi$ is a root of a polynomial with integer coefficients in the first place, and then it becomes a matter of algebra/number theory. – Ted Shifrin Aug 03 '23 at 05:22
  • @TedShifrin is that something I will learn in undergraduate abstract algebra? (I take it this coming semester) – David Raveh Aug 03 '23 at 05:27
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    Not likely in a first course, no. But if you keep going to Galois theory and algebraic number theory, yes. – Ted Shifrin Aug 03 '23 at 06:31
  • "Clearly, $\theta$ can't be rational." Yes, it can – it can be zero. But there aren't any general ways to tell, given $\theta$, whether $\theta/\pi$ is rational. For example, no one knows whether $e/\pi$ is rational. There are some unconvincing reasons for thinking that $\zeta(3)/\pi^3$ is rational, where $\zeta(3)=\sum_1^{\infty}n^{-3}$, but no one has found a proof that it isn't (or that it is). These are, in general, hard, hard questions. – Gerry Myerson Aug 03 '23 at 07:15

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Hint A number $w \in \Bbb C$ satisfies $e^w = 1$ iff $w \in 2 \pi i \Bbb Z$. Now, suppose that $e^{i \theta}$ satisfies $(e^{i \theta})^n = 1$.

Travis Willse
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  • I think I didn't communicate well what I am looking for; I have edited my question so that it is clearer (hopefully). – David Raveh Aug 03 '23 at 03:55
  • In that case I'm still unclear what you're asking. How is the $\theta$ you're given represented? Are you essentially asking, e.g., "How do you show that $\frac{3}{5} + i\frac{4}{5}$ isn't a root of unity without using the language of ring theory?" – Travis Willse Aug 03 '23 at 04:22
  • I have edited to explain that I don't have a specific $\theta$. I am looking for somewhat general methods that can work on some class of values. If you could give the ring theory method (or some other method) and show that it works on a large class of values, that would be good. – David Raveh Aug 03 '23 at 04:59
  • Are you effectively asking, How can you determine whether a given real number $\alpha$ is rational? Like others have indicated in comments under the original post, the answer to that depends on how the number is specified. – Travis Willse Aug 03 '23 at 05:03
  • "how the number is specified" please explain what this means. Do you mean like $a+bi$? Sure, that is one way it can be written... – David Raveh Aug 03 '23 at 05:11
  • The question you've linked shows how to handle at least some numbers of the form $\arctan \frac{p}{q}$, with $p \in \Bbb Z$, $q \in \Bbb Z \setminus {0}$, but only countably many numbers are arctangents of rational numbers. In general it can be difficult to determine whether a given number is rational, and so which techniques apply depends on how the number is given to you. For example, Apery's constant, $\zeta(3) = \sum_{n=1}^\infty n^{-3}$, was only shown to be irrational in the 1970s (a.f.a.I.k. no one expected it to be rational). – Travis Willse Aug 03 '23 at 19:45