Artin-Schreier extensions and Galois module structure. (Q1403938)

Let \(k=\mathbb F((T))\) be the field of Laurent series in \(T\) over the finite field \(\mathbb F\) of characteristic \(p>0\) and \(L\) an Artin-Schreier extension of \(k\) of degree \(p\) such that the valuation ring \(O_L\) of \(L\) is wildly ramified over \(O_k =\mathbb F[[T]]\). The author gives an \(O_k\)-basis for the associated order \(A(L/k) = \{x\in k[\text{Gal}(L/k)] \mid xO_L \subset O_L\}\) and a criterion for \(O_L \simeq A(L/k)\).