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.
It is well known that if D is a Dedekind domain with quotient fieldK and if T is any Hausdorff nondiscrete field topology on K for which the open D-submodules of K form a fundamental system of neighborhoods of zero, then T is the supremum of a family of /?-adic topologies. We show that if the class number of K over D is finite and if T is any Hausdorff nondiscrete field topology on K for which D is a bounded set, then T is the supremum of a family of /?-adic topologies. We then investigate the problem of extending a locally bounded topology from D to a locally bounded topology on K. The extendable topologies on D for which there exists a nonzero topological nilpotent and for which D is a bounded set are characterized. Moreover it is shown that the topology of a locally compact principal ideal domain A extends to a ring topology on the quotient field of A if and only if A is compact.
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