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We study necessary and sufficient conditions for a valued field $\KF$ withvalue group $G$ and residue field $\kf$ (with char $\KF$ = char $\kf$) to admita 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 ofa family ({\it tower of complements}) of $\kf$-subspaces of $\KF$ which arecomplements of the (possibly fractional) ideals of the valuation ring. If $\KF$is a Henselian field of characteristic 0 or, more generally, an algebraicallymaximal Kaplansky field, we give an intrinsic construction of such a familywhich does not rely on a given truncation closed embedding. We also show thattowers of complements and truncation closed embeddings can be extended from anarbitrary field to at least one of its maximal immediate extensions.

Towers of complements to valuation rings and truncation closed embeddings of valued fields