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I know that every linear algebraic group is rational over algebraically closed fields. To what extent is that true for other fields? For example: is $GL_n(\mathbb{Q}_p)$ a rational variety? Are there special cases where linear algebraic groups over non algebraically closed fields are known to be rational?

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    $\begingroup$ $GL_n$ is certainly a rational variety (over any field), since it's birational to $M_n$ which is affine $n^2$ space. $\endgroup$
    – Abhinav Kumar
    Commented Jun 16, 2013 at 3:05
  • $\begingroup$ What is your definition of rational? $\endgroup$
    – Keerthi Madapusi
    Commented Jun 16, 2013 at 4:08
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    $\begingroup$ More generally, any split reductive algebraic group is rational, as follows from the existence of the big cell (an open dense set isomorphic to the product of a Borel subgroup (isomorphic to a variety to a product of affine lines and affine lines minus zero) and the unipotent radical of the opposite Borel (isomorphic to affine space). $\endgroup$
    – Geordie Williamson
    Commented Jun 16, 2013 at 4:12
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    $\begingroup$ The set of rational points has no "algebro-geometric structure". For example, ${\rm{GL}}_n(\mathbf{Q}_p)$ is a group, or even $p$-adic analytic manifold, but it is not a variety. The $\mathbf{Q}_p$-variety ${\rm{GL}}_n$ (denoted $({\rm{GL}}_n)_{\mathbf{Q}_p}$ if you wish) is what is rational, not its set of $\mathbf{Q}_p$-points. Please be clear with the notation. See the paper "Rationality problem for semisimple group varieties" by Chernousov and Platonov for the non-split case, especially the Main Theorem in section 1 (and type-A counterexamples in rank $> 2$, due to Merkurjev and Rost). $\endgroup$
    – user30180
    Commented Jun 16, 2013 at 6:01
  • $\begingroup$ The question seems interesting but the title should maybe be changed. It is obvious that $GL_n$ is rational over any field, as Abhinav Kumar pointed out, but there are linear groups which are not rational (although they are geometrically rational), so the question is probably about these ones. And it is true, as ayanta pointed out, that an algebraic group is not only its set of rational points. $\endgroup$
    – Jérémy Blanc
    Commented Jun 16, 2013 at 9:22

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