Roots of Unity

The -th root of unity is a number such that ; we often call roots of unity the set .


The -th root of unity is a number such that . When we use the plural, -th roots of unity, we refer to the set .

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 which is in fact the 4-th root of unity in the set of complex numbers (you can check this by computing ). The associated set would be .

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 . For example let’s consider the field with prime characteristic . The elements of this field are , , , , , and . Let’s look at the powers of :

So is the -rd root of unity with the corresponding subgroup of order 3 being