It is well known that if 11 II is a norm on the field F( X) of rational functions over a field F for which F is bounded, then II II is equivalent to the supremum of a finite family of absolute values on F(X), each of which is improper on F. Moreover, 11 || is equivalent to an absolute value if and only if the completion of F(X) for || II is a field. We show that the analogous characterization of norms on F(X) for which F is discrete is impossible by constructing for each infinite field F, a norm II II on F(X) such that Fis discrete, || X\\ < 1, the completion of F(X) for || || is a field, but 11 || is not equivalent to the supremum of finitely many absolute values.