  For important cases of algebraic extensions of valued fields, we develop
presentations of the associated K\"ahler differentials of the extensions of
their valuation rings. We compute their annihilators as well as the associated
Dedekind differentials. We then apply the results to Galois defect extensions
of prime degree. Defects can appear in finite extensions of valued fields of
positive residue characteristic and are serious obstructions to several
problems in positive characteristic. A classification of defects (dependent
vs.\ independent) has been introduced by the second and the third author. It
has been shown that perfectoid fields and deeply ramified fields only admit
extensions with independent defect. We give several characterizations of
independent defect, using ramification ideals, K\"ahler differentials and
traces of the maximal ideals of valuation rings. All of our results are for
arbitrary valuations; in particular, we have no restrictions on their rank or
value groups.
