Author’s abstract: Let \(k\) be a perfect field such that \(\overline {k}\) is solvable over \(k\). We show that a smooth, affine, factorial surface birationally dominated by affine 2-space \(\mathbb{A}^2_k\) is geometrically factorial and hence isomorphic to \(\mathbb{A}^2_k\). The result is useful in the study of subalgebras of polynomial algebras. The condition of solvability would be unnecessary if a question we pose on integral representations of finite groups has a positive answer.