On the number of locally bounded field topologies

Abstract:

Kiltinen has proven that there exist ${2^{\left | F \right |}}$ first countable, locally bounded field topologies (the maximum number possible) on a field $F$ of infinite transcendence degree over its prime subfield. We consider those fields $F$ of countable transcendence degree over its prime subfield $E$. In particular it is shown that if the characteristic of $F$ is zero and the transcendence degree of $F$ over $E$ is nonzero or if $F$ is a field of prime characteristic and the transcendence degree of $F$ over $E$ is greater than one, then there exist ${2^{\left | F \right |}}$ normable, locally bounded field topologies on $F$.