Tensor products of valued fields

TL;DR

This paper proves that the tensor product valuation on the tensor product of valued field extensions over an algebraically closed valued field is multiplicative, leveraging quantifier elimination in the theory of algebraically closed valued fields.

Contribution

It provides a concise proof of multiplicativity of tensor product valuations in valued fields using model theory techniques.

Findings

Tensor product valuation is multiplicative over algebraically closed valued fields.

Quantifier elimination in ACVF is used to establish valuation properties.

The result applies to valued extensions with real-valued valuations.

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.