Do all diagonalizable matrices $A\in M_n(\mathbb C)$ have square root in $\mathbb C$?

Question

Let $A\in M_n(\mathbb C)$ be diagonalizable matrix and $\mathbb F = \mathbb C$.

Does it mean this matrix has root? I think this is correct, and thought on proving it with the fact that every polynomial has root in the complex numbers field, but it doesn't work well.

Am I wrong? How am I supposed to approach this question?

Answer 1

Hint: first construct a root for a matrix $\mathrm{diag}(\lambda_1, \dots, \lambda_n)$. Then use this to construct a root of $$M = P\, \mathrm{diag}(\lambda_1, \dots, \lambda_n) P^{-1}$$