Wild cyclic-by-tame extensions

TL;DR

This paper characterizes the ramification filtrations of wildly ramified Galois extensions of local fields with semi-direct product G, and constructs a parameter space with explicit equations for certain wild cyclic extensions.

Contribution

It provides necessary and sufficient conditions for ramification filtrations and introduces a parameter space for G-Galois extensions with explicit equations for degree p^3.

Findings

Conditions for ramification filtrations in wildly ramified extensions

Existence of a parameter space depending only on the ramification filtration

Explicit equations for wild cyclic extensions of degree p^3

Abstract

Suppose G is a semi-direct product of the form Z/p^n \rtimes Z/m where p is prime and m is relatively prime to p. Suppose K is a local field of characteristic p > 0. 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 whose dimension depends only on the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p^3.