Norms on \(F(X)\) (Q1167216)

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