Let $\gcd(p, q) = 1$ and $Y=\{(t^p, t^q) \in \mathbb C^2 \}$. Determine the ideal $I(Y)$.
Definition. The ideal of $X$ is defined as
$$I(X)=\{f\in \Bbb C[x,y]:f(x,y)=0, \forall (x,y)\in X\}.$$
It is clear that $I(Y)$ is the kernel of the homomorphism $ \phi :\mathbb C[x,y] \to \mathbb C[t]$ defined by $ x \to t^p $ and $ y \to t^q$. Further it is clear that $(x^q-y^p)\subseteq\ker\phi$. But I'm unable in showing that $\ker(\phi)$ is contained in $(x^q-y^p)$. Any hints/ideas ?