Wild ramification in a family of low-degree extensions arising from iteration

TL;DR

This paper investigates wild ramification phenomena in a specific family of quadratic iterated extensions, revealing complex factorization patterns and ramification behavior depending on a parameter c.

Contribution

It provides a detailed analysis of wild ramification in iterated quadratic extensions, including explicit factorization and ramification group descriptions.

Findings

2 ramifies wildly in all cases except c=0

Factorization of (2) varies complexly with c

Higher ramification groups are described in some cases

Abstract

This article gives a first look at wild ramification in a family of iterated extensions. For integer values of c, we consider the splitting field of $(x^2 + c)^2 + c$, the second iterate of $x^2 + c$. We give complete information on the factorization of the ideal (2) as c varies, and find a surprisingly complicated dependence of this factorization on the parameter c. We show that 2 ramifies (necessarily wildly) in all these extensions except when c = 0, and we describe the higher ramification groups in some totally ramified cases.