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

Apply polynomial division in $F[x]$:Evaluation at $T$ gives $r(T)=f(T)-q(T)m_T(T)=0$. A nonzero remainder could be rescaled to a monic polynomial annihilating $T$ of smaller degree, contradicting the defining minimality. Hence $r=0$ andIn particular, the annihilating polynomials form the principal ideal $(m_T)$.