Let G be an affine algebraic group defined over k, with k a field of characteristic zero. Suppose G has only one single k-point, can we conclude that G does not have more points?

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non-unipotentsmooth affine groups of positive dimension overanyinfinite field $k$: by Grothendieck, such groups always have a non-trivial $k$-torus, and those are unirational, QED. A variant works over any infinite perfect field in the unipotent case. But over imperfect fields it can fail: over $k(t)$ for $k$ of char. $p > 2$, take $G = {y^p = x - t x^p}$. – BCnrd Jul 14 '10 at 23:05