Rationality of Some Projective Linear Actions

Let K be any field which may not be algebraically closed, V a finite-dimensional vector space over K, σ GL(V) where the order of σ can be finite or infinite. . If dimK V ≤ 3, then both K(V) σ and K( (V)) σ are rational ( = purely transcendental) over K. Similar results hold for a cyclic affine action.