Definition 1

Let $A$ and $A'$ be two valuation rings of the same field $K$. $A$ and $A'$ are called independent if $K$ is the ring generated by $A$ and $A'$. Two valuations on $K$ are called independent if their rings are independent and dependent otherwise.

An improper valuation on $K$ is independent of every valuation on $K$. For two valuations of height 1 on $K$ to be independent, it is necessary and sufficient that they be not equivalent (§ 4, no. 5, Proposition 6 (c)).