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I seek a proof that xn -1 = 0 has a primitive root.

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I don't know what you are looking for. All the roots are trivially expressible, and the proof is that they satisfy the equation.

$x_k=e^{\frac{2\pi ik}{n}},k=[0,...,n-1]$

Edited by mathematic
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Try induction on $x^2-1=0$ which has roots $\sqrt{1}$. Clearly $-1$ is primitive since it is not a root for $x^1-1=0$

i.e. assume for your induction hypothesis that the roots for $x^{n-1}-1=0$ are $^{n-1}\sqrt{1}$

BUT beware of multiplicities!

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Thanks for responses. I was thinking about a proof which did not require the explicit identification of a root, but I accept that

cos 2pi/n + i sin 2pi/n is indeed a primitive root (of xn = 1).

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Looking at my original post (exponential form), it is easy to see that for every n, the roots for k=1 and k=n-1 are primitive. In general roots where k/n is in lowest terms (as a fraction), kth root is primitve.

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