Suppose $k$ is algebraically closed, $A$, $B$ are $k$-algebras and $A$ is an affine $k$-algebra. It is known that then $A\otimes_k B$ is a domain if $A$ and $B$ are domains. This can be found in Milne's Algebraic Geometry notes as Proposition 4.15(b). I do not see where the assumption $k$ algebraically closed is used.
He gives an example that the above is not true if $k$ is not algebraically closed. But i dont see where this assumption is being used in the proof.
I think that for an affine $k$-algebra the Jacobson radical is the nilradical, so here we do not need $k$ to be algebraically closed.