We let $F$ be a non-archimedean local field (say with finite residue field). Consider Galois extensions $E$ of $F$ in a fixed separable closure $\bar{F}$ of $F$. If $E/F$ is finite we let $u(E/F)$ be the highest index $u\geq 0$ for which $G^u \neq 1$ (if $E = F$, we take $u =0$). One can find this index for example in Serre Local Fields VI-§2.

Suppose we extend this notion to infinite Galois extensions $E/F$ by taking the supremum over the indices of all finite subextensions $E'/F$. Does this notion behave well, for example with taking compositum fields (that is, $u(E/F) = \sup_{E'}\, u(E'/F))$?

Thanks in advance, AYK