The $n$-th roots of a complex number and the $n$ distinct roots of unity for every $n\ge1$

Write $\iota_{\mathbb R}:\mathbb N\to\mathbb R$ for the canonical-natural map of The canonical natural $\iota(n) = n \cdot 1_F$ of a field. If $n\in\mathbb N$ with $n\ge1$ and $z=re^{i\theta}\ne0$, its $n$ distinct roots are $r^{1/n}\exp\!\left(i\frac{\theta+2\pi\iota_{\mathbb R}(k)}{\iota_{\mathbb R}(n)}\right),\qquad k\in\mathbb N,\quad 0\le k<n.$ For $z=0$, the only $n$th root is $0$. Thus the $n$th roots of unity are precisely $\exp\!\left(i\frac{2\pi\iota_{\mathbb R}(k)}{\iota_{\mathbb R}(n)}\right)$ for $k\in\mathbb N$ with $0\le k<n$. The conventions and prerequisite facts used below are recorded in Every nonzero complex number has a unique polar form $r(\cos\theta+i\sin\theta)$ with $r>0$ and $-\pi<\theta\le\pi$, Complex de Moivre formula for every integer exponent, $\ker(\exp)=2\pi i\mathbb Z$, and $\exp z=\exp w$ exactly when $z-w\in2\pi i\mathbb Z$, Existence and uniqueness of $n$-th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$, The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Integer powers in the complex field.