3
$\begingroup$

In a question here, the solution given states that $$\zeta=\cos{(\pi/8)}+i\sin{(\pi/8)}$$ is a primitive 8th root of unity. I was under the impression that the primitive roots of unity were given my $$\zeta_n=\cos{(2\pi/n)}+i\sin{(2\pi/n)}$$ which in case wouldn't it be a primitive 16th root of unity? I might have my language messed up here, so I was just looking for clarification.

$\endgroup$
4
  • 1
    $\begingroup$ I agree with you. $\cos(\pi/8)+i\sin(\pi/8)$ is a primitive sixteenth root of unity. $\endgroup$ – Jack D'Aurizio Jun 12 '15 at 13:50
  • $\begingroup$ I was lookiing up ways to determine how to find minimal polynomials for sin(k) where k is in degrees, so I came across that answer and the language confused me for a second. I haven't even gone forward to determine whether the question is right or wrong now, but thank you for clarifying. $\endgroup$ – Iceman Jun 12 '15 at 13:54
  • 1
    $\begingroup$ If $n\gt 2$, there are several primitive roots of unity. A complete list is given by the numbers $\cos(2k\pi/n)+i\sin(2k\pi/n)$, where $k$ ranges over all integers $k$ such that $1\le k\le n-1$ and $k$ and $n$ are relatively prime. The number $\cos(\pi/8))+i\sin(\pi/8)$ is not one of them. $\endgroup$ – André Nicolas Jun 12 '15 at 14:32
  • $\begingroup$ That is what I thought, i was under the assumption of the coprimality, but just referenced the first for the problem mentioned. Thanks for the clarification, @AndréNicolas $\endgroup$ – Iceman Jun 12 '15 at 14:45
4
$\begingroup$

Let me illustrate in case of $4^{th}$ roots of unity. Well $x^4=1$ has 4 roots $1, i, -i , -1$ out of these 4 $i \; \mbox{and} -i$ have a special property that $i^k \neq 1$ if $k<4$ that means they are strictly fourth roots of unity and no smaller power of these numbers is 1.

$\endgroup$
1
  • $\begingroup$ Thanks again. I understood the idea of primitive, but your language was clear. $\endgroup$ – Iceman Jun 12 '15 at 18:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.