Isomorphism between Roots of Unity under Multiplication and Integers under Modulo Addition

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\left({R_n, \times}\right)$ be the complex $n$th roots of unity under complex multiplication.

Let $\left({\Z_n, +_n}\right)$ be the integers modulo $n$ under modulo addition.

Then $\left({R_n, \times}\right)$ and $\left({\Z_n, +_n}\right)$ are isomorphic algebraic structures.

Proof
The set of integers modulo $n$ is the set exemplified by the integers:
 * $\Z_n = \left\{ {0, 1, \ldots, n - 1}\right\}$

The complex $n$th roots of unity is the set:
 * $R_n = \left\{ {z \in \C: z^n = 1}\right\}$

From Roots of Unity:
 * $R_n = \left\{ {1, e^{\theta / n}, e^{2 \theta / n}, \ldots, e^{\left({n - 1}\right) \theta / n} }\right\}$

where $\theta = 2 i \pi$.

Let $z, w, \in R_n$.

Then:
 * $\left({z w}\right)^n = z^n w^n = 1$

and so $z w \in R_n$.

Thus $\left({R_n, \times}\right)$ is a closed algebraic structure.

Consider the mapping $f: \Z_n \to R_n$ defined as:
 * $\forall r \in \Z_n: f \left({r}\right) = e^{r \theta / n}$

which can be seen to be a bijection by inspection.

Let $j, k \in \Z_n$.

Then:

Thus $f$ is an isomorphism.