# Roots of Unity under Multiplication form Cyclic Group

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## Theorem

Let $n \in \Z$ be an integer such that $n > 0$.

The $n$th complex roots of unity under the operation of multiplication form the cyclic group which is isomorphic to $C_n$.

## Proof

From Complex Roots of Unity in Exponential Form:

- $U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$

where $U_n$ is the set of complex $n$th roots of unity.

Let $\omega = e^{2 i \pi / n}$.

Then we have:

- $U_n = \set {\omega^k: k \in \N_n}$

that is:

- $U_n = \set {\omega^0, \omega^1, \omega^2, \ldots, \omega^{n - 1} }$

Let $\omega^a, \omega^b \in U_n$.

Then $\omega^a \omega^b = \omega^{a + b} \in U_n$.

Either $a + b < n$, in which case $\omega^{a + b} \in U_n$, or $a + b \ge n$, in which case:

\(\ds \omega^a \omega^b\) | \(=\) | \(\ds \omega^{a + b}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \omega^{n + t}\) | for some $t < n$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \omega^n \omega^t\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \omega^t\) | as $\omega^n = 1$ |

So $U_n$ is closed under multiplication.

We have that $\omega_0 = 1$ is the identity and that $\omega^{n - t}$ is the inverse of $\omega^t$.

Finally we note that $U_n$ is generated by $\omega$.

Hence the result, by definition of cyclic group, and from Cyclic Groups of Same Order are Isomorphic:

- $U_n = \gen \omega \cong C_n$.

$\blacksquare$

## Also see

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 9$: Cyclic Groups: Example $1$ - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.09$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 44$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Exercise $(1)$ - 1992: William A. Adkins and Steven H. Weintraub:
*Algebra: An Approach via Module Theory*... (previous) ... (next): $\S 1.1$: Example $4$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.6$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $4$: Subgroups: Example $4.8$