Appears in collection : 2018 - T1 - WS2 - Model theory and valued fields

My talk will be devoted to a basic theory of extensions of complete real-valued fields L/K. Naturally, one says that L is topologically-algebraically generated over K by a subset S if L lies in the completion of the algebraic closure of K(S). One can then define topological analogues of algebraic independence, transcendence degree, etc. These notions behave much weirder than their algebraic analogues. For example, there exist non-invertible continuous K-endomorphisms of the completed algebraic closure of K(x). In my talk, I will tell which part of the algebraic theory of transcendental extensions extends to the topological setting, and which part breaks down.