If a linear operator $T$ has minimal, characteristic polynomial $q^k$ for $q$...

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math.stackexchange.com/a/1090868/1048081
Let $T$ be a linear transformation on a finite dimensional vector space $V$ over the field $F$. Let $p_T$ be the minimal polynomial of $T$ and $f_T$ be the characteristic polynomial of $T$. If $p_T = f_T = q^k$ for some irreducible $q$ with $k > 1$, show that no nonzero proper $T$-invariant subspace can have a $T$-invariant complement.