Continuability of cyclic extensions of complete discrete valuation fields

For a complete discrete valuation field $K$ of characteristic 0 with the residue field of characteristic $p>0$ consider the embedding problem of a given cyclic extension $M/K$ of degree $p$ into a cyclic extension of degree $p^n$ for various $n$. Let $c(M/K)$ be the maximal $n$ such that this embedding problem has a solution. In this paper we consider relations between $c(M/K)$ and $c(LM/L)$ where $L/K$ is a given extension linearly disjoint with $M/K$.