I have some confusion on this answer
It is written that
$F[x,y]/(y^2-x) \cong F[u^2,u] = F[u]$
Or consider $ \phi: F[x,y] \to F[u]$ given by $\phi(x)=u^2, \phi(y)=u$, and prove that $\ker \phi = (y^2-x)$.
My Question : How to show that $ \phi: F[x,y] \to F[u]$ given by $\phi(x)=u^2, \phi(y)=u$ is homomorphism
My thinking : I think $ \phi$ is not homomorphism because
$$\phi(xy) \neq \phi(x) \phi(y)=u^2.u=u^3$$
But $\phi(xy)=u^2$ or $\phi(xy)=u $