Let $T$ be a linear map on a finite dimensional vector space $V$ over an arbitrary field. Show that the following 3 statements are equivalent.
(1) The minimal polynomial of $T$ is a product of distinct irreducibles.
(2) Each $T$-invariant subspace is admissible. (An invariant subspace $W$ is admissible if for any polynomial $f$ and $v \in V$ such that $f(T)v \in W$, then there exists $w \in W$ such that $f(T)w=f(T)v$.)
(3) For each $T$-invariant subspace, there exists a $T$-invariant complement.
The equivalence between (2) and (3) is a result in Hoffman and Kunze p.232. I am trying to show the equivalence between (1) and (3). However I don't know how to establish either direction. The problem is easier in complex case because the minimal polynomial is always a product of linear factors. So in the real case it requires just a little more work. but in general field I don't know how to proceed in either direction, any hint/help is appreciated.