The following proposition is proved in EGA. Since EGA is somewhat intimidating(and it is written in French), I'm looking for a more readable proof.
EGA IV-3 (8.4.5) Let $X$ be a quasi-compact scheme over a field $k$. $X$ is geometrically connected if and only if $X\times_k K$ is connected for every finite separable extension field $K$ of $k$.