Wild ramification in a family of low-degree extensions arising from iteration. (English) Zbl 1499.11331
Summary: This article gives a first look at wild ramification in a family of iterated extensions. For \(c\in\mathbb Z\) 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.
MSC:
| 11R21 | Other number fields |
| 11R32 | Galois theory |
| 37P05 | Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps |
| 11R09 | Polynomials (irreducibility, etc.) |
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