It is well known that if ∥⋅⋅∥
is a norm on the field F(X) of rational functions over a field F for which F is
bounded, then ∥⋅⋅∥ is equivalent to the supremum of a finite family of absolute
values on F(X), each of which is improper on F. Moreover, ∥⋅⋅∥ is equivalent to an
absolute value if and only if the completion of F(X) for ∥⋅⋅∥ 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 ∥⋅⋅∥ on
F(X) such that F is discrete, ∥X∥ < 1, the completion of F(X) for ∥⋅⋅∥ is a
field, but ∥⋅⋅∥ is not equivalent to the supremum of finitely many absolute
values.
