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If a question states that $\omega_n$ is THE primitive complex root of unity of order $n$, then does this always imply that $\omega_n$ = $e^{2\pi i/n}$, i.e. primitive roots of unity of order $n$ are those complex numbers such that $\omega_n = e^{2\pi ij/n}$ for which $j$ is such that $0 < j < n$ and $gcd(j,n)$= 1, but if a question asks for THE primitive root, can it be taken as the case $j=1$?

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    $\begingroup$ as I know there is no "the" primitive root, just that it should not be $1$. $\endgroup$ – King Tut Apr 7 '18 at 7:29
  • $\begingroup$ @KingTut there are other roots of unity that are not primitive and not 1 though..? $\endgroup$ – dimebucker Apr 7 '18 at 7:39
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    $\begingroup$ A question shouldn't ask for the primitive $n$th root of unity (unless $n$ is 1 or 2). If it does, the person asking the question had better have previously given a precise definition, so others can know what she means. $\endgroup$ – Gerry Myerson Apr 7 '18 at 10:59
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The equation amounts to solutions for $\omega^n=1$.

The solution for this is $\omega=\exp{(2\pi i j/n)}$.

The solution is primitive when the $\gcd(j,n)=1$, which means that the first power to equal 1, is when n divides the power.

The solution is THE primitive exactly when $j=1$.

The reason for this, is that isomorphisms are used to project the primitive (which is taken as the identity in isomorphisms), onto other solutions. The same situation occurs in

$x^2=4$. Thus, $2$ and $-2$ are square roots of $4$, but $x=2$ is THE square root of $4$.

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  • $\begingroup$ What is your source for this, wendy? It's not a distinction I can recall ever seeing anyone make. $\endgroup$ – Gerry Myerson Apr 7 '18 at 11:00
  • $\begingroup$ @GerryMyerson I am not sure what you mean by source. It's a distinction one makes to deal with polygonal isomorphism. $\endgroup$ – wendy.krieger Apr 7 '18 at 11:06
  • $\begingroup$ I mean, who exactly makes that distinction? Where does it say (other than in your answer) that $e^{2\pi i/n}$ is the primitive root? Can you cite a textbook, or a paper? $\endgroup$ – Gerry Myerson Apr 7 '18 at 12:57
  • $\begingroup$ The people who use cyclotomic numbers as a basis of a number system make this distinction. For example, a particular transform is to replace $\exp{2\pi i j/n}$ by $\exp{2\pi i jk/n}$, which transforms a polygon into one of its transforms. It's the solution to Coxeter's isomorphism problem. But i ultimately rely on pictures I saw in Gauss's "Discussion on Arithmetic" i saw when i was 12 or something. $\endgroup$ – wendy.krieger Apr 7 '18 at 13:32
  • $\begingroup$ You and I must have different editions of the Disquisitiones – mine doesn't have any pictures in it. $\endgroup$ – Gerry Myerson Apr 7 '18 at 21:51

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