Question: Given is a regular polygon with n edges, which has a corner point located on the positive real axis. Show that for all n corner points is valid $z_k$ gilt: $z_k^n=1$
Above is an assignment question(for polar coordinate system), I do not know what is the exact meaning conveyed here since some German math term is used. please help understand and solve.
EDIT:
From this, General form, Since a vertice lies in positive real axis, $$ z_k = r_ke^{iθ_k}, k= 0, 1, ...(n-1);\\ \begin{align} & => r_k[cos(k.2π)+isin(k.2π)] \\ & = r_k\\ \end{align} $$ How to say this is true for a regular polygon without unity r and not lying in center ?, Is it possible ?
Embed a regular polygon with $n$ edges in the complex plane (with one vertex on the $x$-axis, i.e. $1$) and denote it's vertices by $z_1, \dots, z_n$, then show for every $1\leq k\leq n$ that $z_k^n=1$.
For example, for the triangle we have that the corners lie on the three points $1$ and $-\frac{1}{2}\pm i\frac{\sqrt{3}}{2}$. This all relates to a concept in complex analysis called the "roots of unity", as you will show through your work on this problem that the vertices of these polygons are the roots of the equation $x^n=1$.
Regarding the edit:
In general, if you wish to center the polygon at $\alpha \in \mathbb{C}$, have radius $r$, and have the root that is normally at $\alpha+1$ rotated by $\theta$ radians about $\alpha$, you would be considering the roots of
$$(x-\alpha)^n=r^ne^{i\theta}.$$