Tensor products of valued fields

Abstract

We give a short argument why the tensor product valuation on $K \otimes_k L$ is multiplicative when $k$ is an algebraically closed valued field and $K$ and $L$ are valued extensions (all valuations being in $\bR$). When the valuation on $k$ is non trivial we use the fact that $ACVF$, the theory of algebraically closed (non trivially) valued fields, has quantifier elimination.