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I am not entirely sure if there is a difference between $K$-rational points on a scheme $X$ over $k$ and $K$-valued points on $X$. Both seem to refer to a $k$-morphism Spec $K \to X$ but the difference is that in $K$-rational, $K/k$ is a field extension whereas in $K$-valued, $K/k$ might just be a containment of rings? Thanks.

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    $\begingroup$ Containment of rings=field extension for fields. The terms are synonomous. And so yes, K-rational and K-points mean the same thing. :) $\endgroup$ Jul 23, 2015 at 12:32

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