Topological transcendence degree

Throughout the paper, an analytic field means a non-archimedean complete real-valued one, and our main objective is to extend to these fields the basic theory of transcendental extensions. 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 asserts that if $\mathrm{char}(k)>0$ then $\widehat{k((t))^a}$ 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 $\widehat{\mathbf{F}_p((t))^a}$ are discussed.