Let $F$ be a field and let $K$ be a field extension of $F$. Suppose that $K$ can be written as the compositum of field extensions $E$ and $L$ of $F$, linearily disjoint over $F$, thus $K$ can be identified with the field of fractions $E \otimes _F L$, an integral domain.
Suppose that we are given a valuation of $E$ over $F$ with residue field $F$, for example, $E$ could be the field of Laurent series over $F$. Can $K$ be given a valuation over $L$ with residue field $L$ which is compatible with this valuation? Is such a valuation unique? Note that the valuation need not be of rank one or archimedean.
This is probably very basic, but I am new to this material.