Let $A=k[X]$ and $f\in A$; then set $B=k[X,Y]/(Y^2-f)$. Find the necessary and sufficient condition that $f$ must satisfy in order for $B$ to be an integral domain. Assume this holds, and write $K$ for the field of fractions of $B$, that is $K=k(X)(\sqrt f)$. For any $\alpha \in K$, write down a polynomial $h(t)\in A[T]$ such that $h(\alpha)=0$. Show that $B$ is normal if and only if $f$ has no square factors. If $B$ is not normal show how to obtain its normalisation in terms of $f$.
For the first part, $(Y^2-f)$ should be a prime ideal, it cannot be $(0)$ since $f \in A$. So, there are two other options for this ideal but I could not find a condition for $f$. For the rest I have some ideas but could not obtain anything useful. Also, there is a hint provided: Write $\alpha=a+b\sqrt f$, where $a,b \in K(X)$.