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

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Theorem

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

Let $\struct {R_n, \times}$ be the complex $n$th roots of unity under complex multiplication.

Let $\struct {\Z_n, +_n}$ be the integers modulo $n$ under modulo addition.


Then $\struct {R_n, \times}$ and $\struct {\Z_n, +_n}$ are isomorphic algebraic structures.


Proof

The set of integers modulo $n$ is the set exemplified by the integers:

$\Z_n = \set {0, 1, \ldots, n - 1}$

The complex $n$th roots of unity is the set:

$R_n = \set {z \in \C: z^n = 1}$

From Complex Roots of Unity in Exponential Form:

$R_n = \set {1, e^{\theta / n}, e^{2 \theta / n}, \ldots, e^{\left({n - 1}\right) \theta / n} }$

where $\theta = 2 i \pi$.


Let $z, w, \in R_n$.

Then:

$\paren {z w}^n = z^n w^n = 1$

and so $z w \in R_n$.

Thus $\struct {R_n, \times}$ is a closed algebraic structure.


Consider the mapping $f: \Z_n \to R_n$ defined as:

$\forall r \in \Z_n: \map f r = e^{r \theta / n}$

which can be seen to be a bijection by inspection.


Let $j, k \in \Z_n$.

Then:

\(\ds \map f j \map f k\) \(=\) \(\ds e^{j \theta / n} e^{k \theta / n}\)
\(\ds \) \(=\) \(\ds e^{j \theta / n + k \theta / n}\)
\(\ds \) \(=\) \(\ds e^{\paren {j + k} \theta / n}\)
\(\ds \) \(=\) \(\ds \map f {j +_n k}\)

Thus $f$ is an isomorphism.

$\blacksquare$


Sources

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