Let $k$ is analgebraically closed field, $K$ contains $k$ and$A$ is a $k$ algebra. $p$ is a prime of A.
I want to proof that the claim that $k$ is algebraically closed, implies that $A/\mathfrak{p} \otimes_k K$ is a domain.
My attempt: $K= k(t_1,..,t_n)$ since $k$ algebraically closed, therefore $A/\mathfrak{p}_i\otimes_k K \cong (A/\mathfrak{p})(t_1,..,t_n)$. (is this correct?)
If yes, then I can observe that since $A/\mathfrak{p}_i$ is a domain, then also $ A/\mathfrak{p}(t_1,..,t_n)$.
Is this proof correct?
Here the background of my question: How do I prove a scheme is connected?