Suppose $X$ is an irreducible nonsingular projective variety over a field $k$ (not necessarily algebraically closed) Let $K$ be a field extension of $k$ ( If $K/k$ is not algebraic, we can assume that the field $k$ is infinite. I don't know if this assumption is useful or not. But if necessary one can assume this). Now can we prove that the base extension $X'$= $X$ x Spec $K$ over Spec $k$ is irreducible?
If $k$ is algebraically closed, then I know that this is true. Otherwise I don't know how to prove?