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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 $k$.

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This is Lemma 0387 in the Stacks project. –  Zhen Lin Dec 25 '12 at 1:12
    
@ZhenLin Thanks. I guess you mean Lemma 28.5.11 of the book version. –  Makoto Kato Dec 25 '12 at 4:03
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