General linear group over algebraically closed field is divisible

Statement

Suppose $K$ is a field that is algebraically closed and $G$ is a general linear group of finite degree $n$ over $K$, i.e., $G=GL(n,K)$. Then, $G$ is a divisible group, i.e., for any $g\in G$ and any positive integer $n$, there exists $x\in G$ (not necessarily unique) such that $x^{n}=g$.

Related facts

• Triangulability theorem

Proof

The idea is to first conjugate to a Jordan canonical form matrix, then take the unique $n^{th}$ root of that.