how to extend the valued field embedding?

Question

let $(K,V)$ and $(L,W)$ be two algebraically closed valued fields,and $E$ be a sub-field of $K$ . for a valued field embedding $f$ from $E$ to $L$, how should I extend it to a valued field embedding from algebraically closure of $E$ to $L$?

Answer 1

This is a crucial step in the proof of quantifier elimination for ACVF, the theory of algebraically closed valued fields. So you can find proofs or at least references in many sources which prove this theorem.

For example: In these notes by Freitag, the statement you want is exactly Lemma 2.5. Freitag gives a short proof, but refers to other references for the proof of a crucial algebraic component of the proof (Theorem 2.6).

In these notes by van den Dries, the statement you want is Corollary 3.14 (but phrased in the language of valuation rings, rather than valuations - see p.25 for the correspondence between valuation rings and valuations). All the algebraic details are in the other results stated on p. 28, whose proofs are given on p. 29 and p. 30.