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

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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 (x² + c)² + c, the second iterate of x² + 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.