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I already asked this question on Math Stack few days ago ( torsors over a separable closure ), but did not receive any answer, so I post it here.

Let $G$ be a smooth linear algebraic group defined over a field $F$, and let $T$ be a $G$-torsor over $F$. I am pretty sure that it is well-known that $T$ has a point over a separable closure of $F$, but I cannot find any reference.

This is equivalent to say that any $G$-torsor splits after a base change to a suitable finite separable extension of $F$.

Any pointers towards a reference or a proof of this fact (or a counter example if its is false !!!) would be really appreciated.

Thank you !

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    $\begingroup$ This is a property of smooth schemes, and groups and torsors have nothing to do with it. See Bosch-Lütkebohmert-Raynaud, Néron Models, cor. 2.2/13. $\endgroup$
    – Matthieu Romagny
    Commented Apr 13, 2018 at 11:27
  • $\begingroup$ and since a $G$-torsor over a smooth algebraic group is smooth, the corollary applies. Thanks a lot! $\endgroup$
    – GreginGre
    Commented Apr 13, 2018 at 11:31

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