An algebraic proof of a cancellation theorem for surfaces

Let k be an algebraically closed field of arbitrary characteristic. We give a self-contained algebraic proof of the following statement: If V is an affine surface over k such that V×k k3, then V k2. This fact, which is due to Fujita, Miyanishi, Sugie, and Russell, solves the Zariski cancellation problem for surfaces. To achieve our proof, we first show that if A is a finitely generated domain with AK(A)=A, then AK(A[x])=A.