TITLE: When are minimal and characteristic polynomials the same?
QUESTION [43 upvotes]: Assume that we are working over a complex space $W$ of dimension $n$. When would an operator on this space have the same characteristic and minimal polynomial? 

I think the easy case is when the operator has $n$ distinct eigenvalues, but what about if it is diagonalizable? Is that sufficient, or can there be cases (with repeated eigvals) when char poly doesn't equal min poly? What are the general conditions when the equality holds? Is it possible to define them without use of determinant? (I am working by Axler and he doesn't like it.)
Thanks.

REPLY [5 votes]: Here is a generalization to principal ideal domains. 
Let $A$ be a principal ideal domain, $p$ an irreducible element of $A$, and $M$ a finitely generated $A$-module annihilated by some power of $p$. 
Then there is a unique nondecreasing tuple $(n_1,\dots,n_k)$ of positive integers such that $M$ is isomorphic to the direct sum of the $A/(p^{n_i})$. 
The characteristic ideal of $M$ is $(p^s)$, where $s$ is the sum of the $n_i$; and the annihilator of $M$ is $(p^{n_k})$. Let $\phi$ be the endomorphism $x\mapsto px$ of $M$. 
The following conditions are clearly equivalent: 

$k=1$, 
$s=n_k$, 
$\text{Ker }\phi\simeq A/(p)$, 
$\text{Coker }\phi\simeq A/(p)$.