Finitely ramified iterated extensions

Authors

W Aitken
F Hajir, University of Massachusetts - AmherstFollow
C Maire

Publication Date

2005

Journal or Book Title

INTERNATIONAL MATHEMATICS RESEARCH NOTICES

Abstract

Let K be a number field, t a parameter, F = K(t), and φ(x)∈ K [x] a polynomial of degree d ≥ 2. The polynomial Φ_n(x,t) = φ^{^n} (x) − t ∈ F[x], where φ^{^ n} = φ ^ φ ^ … ^ φ is the n-fold iterate of φ, is irreducible over F; we give a formula for its discriminant. Let F be the field obtained by adjoining to F all roots (in a fixed ) of Φ_n(x,t) for all n ≥ 1; its Galois group Gal(F_φ/F) is the iterated monodromy group of φ. The iterated extension F_φ is finitely ramified over F if and only if φ is postcritically finite. We show that, moreover, for post-critically finite φ, every specialization of F_φ/F at t = t₀ ∈ K is finitely ramified over K, pointing to the possibility of studying Galois groups of number fields with restricted ramification via tree representations associated to iterated monodromy groups of postcritically finite polynomials. We discuss the wildness of ramification in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogène number fields that arise from the construction.

Comments

This is the pre-published version harvested from ArXiv. The published version is located at http://imrn.oxfordjournals.org/content/2005/14/855.short

Pages

855-880

Issue

14

Recommended Citation

Aitken, W; Hajir, F; and Maire, C, "Finitely ramified iterated extensions" (2005). INTERNATIONAL MATHEMATICS RESEARCH NOTICES. 413.
Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/413