Wild tame-by-cyclic extensions

Suppose G is a semi-direct product of the form Z/pn⋊Z/m where p is prime and m is relatively prime to p. Suppose K is a complete discrete valuation field of characteristic p>0 with algebraically closed residue field. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration, and we calculate its dimension in terms of the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p3.