Roots Of Unity

Formal Definition

The elements of the set $U_n=\{z\in\mathbb{C}|z^n=1\}$ are called the $n^{th}$ roots of unity.The elements of this set are numbers $\cos（m\frac{2\pi}{n}）+i\sin（m\frac{2\pi}{n}）$ for $m=0,1,2,……,n-1$.They all have absolute value 1, so $U_n\subset U$.If we let $\xi=\cos（m\frac{2\pi}{n}$, then these $n^{th}$ roots of unity can be written as $1=\xi^o,\xi^1,\xi^2,\xi^3,……,\xi^{n-1}$.Because $\xi^n=1$,these $n$ powers of $\xi$ are closed under multiplication.

Informal Definition

Similar to addition modulo n with $\xi$ equal to some number in $U_n$.

Example(s)

Reference HW1 Problem 13 and HW1 Problem 14.

Non-example(s)

The solution of the equation $x+5=3$ in $\mathcal{Z}_8$ is $x=6$,because $5+_8 6=11-8=3$ is not an example because it doesn't have any reference to $xi$ equaling a number in $U_n$.