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Let $K/F$ be a field extension, and let $X$ and $Y$ be affine varieties over $F$. (E.g. they are defined by polynomials over $F$.) Suppose $X$ contains $F$-points. Now view $X$ and $Y$ as $K$-varieties and suppose there exists a proper morphism $f:X\to Y$ as $K$-varieties. (E.g. $f$ is defined by polynomials over $K$ instead of $F$.) My question is then, since $X$ contains $F$-points, is it the case that $Y$ also contains $F$-points? Thank you.

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    $\begingroup$ What happens if $X=Spec(\mathbb{R})$, $Y=Spec(\mathbb{R}[x]/(x^2+1))$ and $f$ maps $X$ to $i\in Y(\mathbb{C})$? $\endgroup$
    – Fan Zheng
    Commented Dec 17, 2016 at 21:58
  • $\begingroup$ @FanZheng Thanks. I was being very naive. $\endgroup$
    – Jimmy
    Commented Dec 17, 2016 at 22:35

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