Sous-corps fermés d'un corps valué (Q796577)

K is a complete non-Archimedean valued field and K(X) denotes a transcendental extension of K. The valuations on K(X), extending the valuation of K are classified. For a fixed valuation on K(X), one makes a valuation on the algebraic closure \(\overline{K(X)}\) of K(X) and \(\overline{K(X)}{\hat{\;}}\) is the completion of \(\overline{K(X)}.\) The main results of the paper are: Theorem 1. If the residue field of K has characteristic 0 then the only algebraically closed and complete fields between K and \(\overline{K(X)}{\hat{\;}}\) are \=K{\^{\ }} and \(\overline{K(X)}{\hat{\;}}\). - Theorem 2. If the residue field of K has positive characteristic then there are infinitely many algebraically closed and complete fields between K and \(\overline{K(X)}{\hat{\;}}.\) A special case of Theorem 2 is: \(\overline{{\mathbb{F}}_ p((T))}{\hat{\;}}\) contains infinitely many algebraically closed and complete subfields. The two theorems have implications for the continuous Galois group of \(\overline{K(X)}{\hat{\;}}\) over K. In the proofs an important rôle is played by the different of an infinite field extension.