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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`$({\rm{GL}}_n)_{\mathbf{Q}_p}$`

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). – user30180 Jun 16 '13 at 6:01