Topological transcendence degree

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