In Hoffman and kunze linear algebra Sec.7.1 , p-228.
If $T$ be a linear operator on a finite dimensional vector space$ V$ over a field $F $.
I can understand how minimal polynomial of a linear operator $ T$ belongs to the ideal $T$-annihilator of $\alpha$ , for every $\alpha \in V,$ as $M (\alpha ;T) $.
But I can't understand how every element of the ideal $T$-annihilator of $\alpha $ divide the minimal polynomial of $T $.
the confusion started from p-202 , sec-6.4 ,
where in the case of $T $-conductor of $\alpha $ into $W $ , $S (\alpha; W) $ . In Hoffman and kunze it's written that " every $T$-conductor divides the minimal polynomial for $T $".