Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2010 / 1 / d

The polynomial $X(X-1)$ annihilates $E$ by idempotence, so the minimal polynomial divides it. If $E=0$, its monic polynomial of smallest positive degree is $X$. If $E\ne 0$, both eigenvalues zero and one occur, so both factors must divide every annihilating polynomial. Therefore