Root of unity

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 $K$. An algebraically closed field contains a primitive root of any order that is relatively prime with its characteristic. In the field of complex numbers, 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

[1]	S. Lang, "Algebra" , Addison-Wesley (1984)