  We study necessary and sufficient conditions for a valued field $\KF$ with
value group $G$ and residue field $\kf$ (with char $\KF$ = char $\kf$) to admit
a truncation closed embedding in the field of generalized power series $\kf((G,
f))$ (with factor set $f$). We show that this is equivalent to the existence of
a family ({\it tower of complements}) of $\kf$-subspaces of $\KF$ which are
complements of the (possibly fractional) ideals of the valuation ring. If $\KF$
is a Henselian field of characteristic 0 or, more generally, an algebraically
maximal Kaplansky field, we give an intrinsic construction of such a family
which does not rely on a given truncation closed embedding. We also show that
towers of complements and truncation closed embeddings can be extended from an
arbitrary field to at least one of its maximal immediate extensions.
