Past exam of the algebra course of the University of Oxford / 2011 / 3 / b / Solution

The universal property of the polynomial ring gives the unique complex algebra homomorphismDistinct monomials $X^{2j}$ are linearly independent, so $\iota$ is injective. Restriction of scalars defines $b\cdot m=\iota (b)m$ for $b\in B$ and $m\in M$; its module axioms follow from those of the original action and the algebra homomorphism. In particular $Y$ acts as the square of the operator defining the $X$ action.