Roots of Unity

The $n$-th root of unity is a number $\omega$ such that ${\omega }^n=1$; we often call roots of unity the set $\left\{1,\omega ,{\omega }^2,\dots ,{\omega }^{n-1}\right\}$.

The $n$-th root of unity is a number $\omega$ such that ${\omega }^n=1$. When we use the plural, $n$-th roots of unity, we refer to the set $\left\{1,\omega ,{\omega }^2,\dots ,{\omega }^{n-1}\right\}$.

Example in the complex numbers

If we were working in a regular (read: not finite) field, the roots of unity would be complex numbers. For example you might already be familiar with the complex number $i$ which is in fact the 4-th root of unity in the set of complex numbers (you can check this by computing $i^4$). The associated set would be $\left\{1,i,-1,-i\right\}$.

Example in a finite field

In a finite field, the definition of “roots of unity” directly maps to that of a multiplicative subgroup of order $n$. For example let’s consider the field ${\mathbb{F}}_7$ with prime characteristic $7$. The elements of this field are $0$, $1$, $2$, $3$, $4$, $5$ and $6$. Let’s look at the powers of $2$:

$$\begin{gathered} 2^1\equiv 2(\text{mod}7) \\ {2}^2\equiv 4(\text{mod}7) \\ {2}^3\equiv 1(\text{mod}7) \end{gathered}$$

So $2$ is the $3$-rd root of unity with the corresponding subgroup of order 3 being $\left\{1,2,4\right\}$.