Given a field $K$, a vector space $V$ over $K$ and a linear map $T:V \longrightarrow V$ of finite dimension with minimal polynomial $\mu(x)\in K[X]$, prove using the primary decomposition of V (modules) that $\mu$ factors completely as a product of different monic linear polynomials if, and only if, $T$ is diagonalizable.
I have tried using linear algebra techniques but primary decomposition results problematic. Any hint will be more than welcome.
Thanks in advance