On fields having the extension property

Abstract

Let T be an intermediate over the field K. We say that K has the Extension Property if every automorphism of K(T) extends an automorphism of K. Fields with this property play a crucial role in the study of homogeneity conditions in spaces of orderings of function fields. In this paper we introduce a new extension property (Generalized Extension Property) and we show it has a better algebraic behaviour. We also rescue most known results about fields with the extension property and determine a new class of fields verifying the generalized extension property.