  Assume that $(L,v)$ is a finite Galois extension of a valued field $(K,v)$.
We give an explicit construction of the valuation ring $\cO_L$ of $L$ as an
$\cO_K$-algebra, and an explicit description of the module of relative K\"ahler
differentials $\Omega_{\cO_L|\cO_K}$ when $L|K$ is a Kummer extension of prime
degree or an Artin-Schreier extension. The case when this extension has
nontrivial defect was solved in a recent paper by the authors with Anna Rzepka.
Using this description, we characterize when $\Omega_{\cO_L|\cO_K}=0$ holds for
an arbitrary finite Galois extension of valued fields. As an application of
these results, we give a simple proof of a theorem of Gabber and Ramero, which
characterizes when a valued field is deeply ramified. We further give a simple
characterization of deeply ramified fields with residue fields of
characteristic $p>0$ in terms of the K\"ahler differentials of Galois
extensions of degree $p$.

Commutative Algebra