Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford / 2013 / 1 / d / Solution

Write $m_1,m_2$ for the minimal polynomials of the two restrictions. Since $p(T)$ vanishes on $U$ and $q(T)$ vanishes on $W$, part (b) givesHence $m_1,m_2$ are coprime. By the direct-sum decomposition, a polynomial annihilates $T$ exactly when it annihilates both restrictions, which happens exactly when it is divisible by both $m_1$ and $m_2$. ConsequentlyIf $m_1=p$ and $m_2=q$, this immediately gives $m_T=pq=f$. Conversely, if $m_T=pq$, the monic polynomials $p/m_1$ and $q/m_2$ have product one, so each is one. ThusThis also handles a zero summand, whose minimal polynomial is one; a nonconstant factor cannot be its minimal polynomial.