On the number of locally bounded field topologies

Abstract

Abstract. Kiltinen has proven that there exist 2|/r| 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|f| normable, locally bounded field topologies on F.