View source for Root of unity

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An element $\zeta$ of a ring $R$ with unity $1$ such that $\zeta^m = 1$ for some $m \ge 1$.  The least such $m$ is the order of $\zeta$.  

A ''primitive'' $m$-th root of unity is an root of unity $\zeta$ of exact order $m$:  $\zeta^m = 1$ and $\zeta^r \neq 1$ for any positive integer $r < m$. The element $\zeta$ generates the [[cyclic group]] $\mu_m$ of roots of unity of order $m$.  For any $k$ that is relatively prime to $m$, the element $\zeta^k$ is also a primitive root. The number of all primitive roots of order $m$ is equal to the value of the [[Euler function]] $\phi(m)$ if $\mathrm{hcf}(m,\mathrm{char}(K)) = 1$.  The primitive roots of unity are the roots of the [[cyclotomic polynomial]] of order $m$.

If the field $K$ contains a primitive root of unity of order $m$, then $m$ is relatively prime to the [[Characteristic of a field|characteristic]] of $K$. An [[algebraically closed field]] contains a primitive root of any order that is relatively prime with its characteristic.  In the field of [[complex number]]s, there are roots of unity of every order: those of order $m$ take the form
$$
\cos \frac{2\pi k}{m} + i \sin   \frac{2\pi k}{m}
$$
where $0 < k < m$ and $k$ is relatively prime to $m$.


====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,   "Algebra" , Addison-Wesley  (1984)</TD></TR>
</table>