Geodesic
Stably rational surfaces over a quasi-finite field
Izvestiya. Mathematics , Tome 83 (2019) no. 3, pp. 521-533

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Let $k$ be a field and $X$ a smooth, projective, stably $k$-rational surface. If $X$ is split by a cyclic extension (for example, if the field $k$ is finite or, more generally, quasi-finite), then the surface $X$ is $k$-rational.
Keywords: rational surfaces, stable rationality, quasi-finite fields, cyclic splitting, Brauer group. 
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     author = {J.-L. Colliot-Th\'el\`ene},
     title = {Stably rational surfaces over a quasi-finite field},
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J.-L. Colliot-Thélène. Stably rational surfaces over a quasi-finite field. Izvestiya. Mathematics , Tome 83 (2019) no. 3, pp. 521-533. http://geodesic.mathdoc.fr/item/IM2_2019_83_3_a4/