Using the fact that the K—automorphism a defined on k(X1,X2…,X8) byis not rational (if k is the field of rational numbers), we see that the K—automorphism β defined on k(X1,X2,X3,X4) byis not rational. We then show that among non—rational purely monomial K—automorphisms on k(X1,X2,…,Xn), this example β is minimal in the sense that neither 8 (the order of β) nor 4 (the number of variables) can be improved upon.