Number of nth roots of unity

Statement

Let $n$ be a positive integer. Then there are exactly $n$ complex $n$th roots of unity.

Remarks

The list of the nth roots of unity has $n$ numbers $e^{2 \pi i k / n}$ with $0 \le k < n$. They are pairwise different, because their arguments are different numbers in $[0, 2\pi)$. A polynomial of degree $n$ has at most $n$ roots in a field, so $X^n - 1$ has no other roots. In other fields there can be fewer: in $\mathbb{R}$, the only $4$th roots of unity are $1$ and $-1$.

Explanations

L1 human

The equation $z^n = 1$ has exactly $n$ complex solutions. Over the real numbers, it has only one or two solutions, so most of the solutions are not real.