Topological transcendence degree

Throughout the paper, an analytic field means a non-archimedean complete real-valued field, and our main objective is to extend basic theory of transcendental extensions these fields. One easily introduces topological analogue transcendence degree, but, surprisingly, it turns out that may behave very badly. For example, particular case theorem Matignon-Reversat, [8, Thèoréme 2] , asserts if char ( k ) > 0 then t ˆ possesses non-invertible continuous -endomorphisms, this implies degree not additive in towers. Nevertheless, we prove some aspects behaves reasonably, show by explicit counter-examples positive results are pretty sharp. Applications types points Berkovich spaces untilts F p discussed.