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If you take a conic through 5 rational points on a quartic curve, then will at least one of the remaining 3 points also be rational ?

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No. Let the conic be the union of lines L_1 and L_2, one of which intersects the quartic Q in four rational points. Now the intersection of L_2 with Q is parametrized by the roots of a quartic equation which can be whatever it wants; in particular it can have one root rational and the rest not.

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    $\begingroup$ @JSE Isn't this a bit too much of an exercise to be answering, instead of suggesting the the OP think about it some more (and maybe give him a hint)? If he wrote down even one or two examples, the answer should be pretty obvious. $\endgroup$ Nov 1, 2011 at 20:42
  • $\begingroup$ That is a good point, but a) OP's other questions suggest this isn't homework, and b) I think "do the degenerate case, it's easier" is a good slogan to promote when you get the chance. $\endgroup$
    – JSE
    Nov 2, 2011 at 1:28
  • $\begingroup$ @JSE: Okay, but he posted two very basic questions at the same time, with no indication that he'd given them any thought or tried to find them in a standard reference. Standard MO practice says this is not good. So I think the right response here would be "what have you tried", instead of spoon-feeding him the answer. Or maybe "try the case where the curve(s) are reducible", but even that seems like too much of a hint. How about "No, but you should figure out why yourself" as an answer? But then again, it's late, I'm tired, so maybe it's just me feeling cranky! $\endgroup$ Nov 2, 2011 at 2:23
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    $\begingroup$ Would it cheer you up if I told you I just used your specialization theorem like 10 minutes ago? $\endgroup$
    – JSE
    Nov 2, 2011 at 2:51

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