Wild tame-by-cyclic extensions

Abstract

Suppose G is a semi-direct product of the form Z / p n ⋊ 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 p 3 .