Exercise 11, § 5

Let K be an algebraically closed field, let $A$ be an invertible matrix over K, and $m > 0$ an integer. Show that, if m is not a multiple of the characteristic of K, then there exists a polynomial $g(X) \in K[X]$ such that $(g(A))^m = A$ (use the Cayley-Hamilton theorem and Ex. 28 of VII, p. 54). Generalise to the equation $f(U) = A$ ($f \in K[X]$, and U an unknown $n \times n$ matrix over K).