General linear group over algebraically closed field is divisible

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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.

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