To be precise, the question is : Let $k$ be a finite field. Let $P_{1}, \ldots, P_{m} \in k^{2}$ be distinct points and let $\lambda_1, \ldots \ldots, \lambda_{m} \in k$; show that there exists $f \in k[x, y]$ such that for all $i$ we have $f\left(P_{i}\right)=\lambda_{i}$.
at first glance this looked like showing an equivalence relation over a projective space, however the method desired seems to be different as it gives the following hint:
for a given $P=(a, b) \in k^{2}$ consider first the following polynomial: $f_{a, b}:= \prod_{v \in k, v \neq a}(X - v)$ * $ \prod_{w \in k, w \neq b}(Y - w)$ . Where is $f$ equal to $0$ and where it is not equal to $0$ ?
im not sure how to use this hint to solve the question? the $f$ given is non zero for $(a,b)$ and zero everywhere else but im not quite sure how that helps. Can anyone point me in the right direction?
thanks