Does every plane curve contain a rational point?

Question

I think the answer is yes, but I can not prove this. Please help.

However, if it is possible to build a pathological curve - without rational points, then even more interesting question arises - which properties of a curve will imply existence of a rational point?

The answer is NO. e.g. $y = x + \pi$ doesn't contain any rational points. — achille hui, Oct 19 '14 at 11:44
@achillehui if we take $x = -\pi$ then this curve contains $0$ which is pretty rational. — Jihad, Oct 19 '14 at 11:46
Not necessarily because $x=\pi $ is a curve which does not contain a rational point. — Jasser, Oct 19 '14 at 11:49
@Jihad A rational point in a plane means a point whose $x$ and $y$ coordinates are both rational. — achille hui, Oct 19 '14 at 11:50
@JohnBentin probably you want to say $x^2+y^2 = \pi^2$. This curve is pretty interesting but doesn't contain any point that have 2 rational coordinates. It's true. — Jihad, Oct 19 '14 at 11:52
@achille hui thanks, indeed it is simple, evident and beautiful counterexample. — Hedgehog, Oct 19 '14 at 11:58

Eclipse Sun · Accepted Answer · 2017-10-25T21:47:58.473

5

The answer is NO. We can show more by only considering straight lines and the fact that rational points are countable.
Choose an arbitrary point $A$ in ${{\mathbb{R}}^{2}}$ whose coordinates are both irrational. The set $L = \left\{ l:A\in l \right\}$ is uncountable, and thus there is no one-to-one mapping from $L$ to ${{\mathbb{Q}}^{2}}$. Hence, there are uncountably many lines going through $A$ which contains no rational points.
If we go further, we can show that ${{\mathbb{R}}^{2}} - {{\mathbb{Q}}^{2}}$ is path-connected.

edited Oct 25 '17 at 21:47

answered Oct 19 '14 at 11:56

Eclipse Sun

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thanks a lot for the answer and for "If we go further", part. – Hedgehog Oct 19 '14 at 12:05

Does every plane curve contain a rational point?

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