Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 4 / 12F / v / Solution

Use the usual convention that a ring homomorphism preserves the identity. Such a map from the quotient ring to $\mathbb{C}$ fixes $\mathbb{Q}$: it fixes integers, and preservation of inverses fixes rational numbers. It is therefore determined by the image $\zeta$ of the class of $x$. The defining relation forces ${\zeta }^3=1$. Conversely, evaluation at any such $\zeta$ annihilates $(x^3-1)$ and hence factors through the quotient. Thus there are exactly three unital homomorphisms, namelyIf the convention permits nonunital ring homomorphisms, add the zero map. These are the only extra possibilities: the image of $1$ is an idempotent in $\mathbb{C}$, hence either zero or one, and image zero forces the whole map to vanish.