Finitely ramified iterated extensions
By W. Aitken, F. Hajir, Christian Maire. 2004
57 citations recorded by the source corpus.
Let K be a number field, t a parameter, F = K(t), and '(x) ∈ K(x) a polyno- mial of degree d ≥ 2. The polynomialn(x, t) = ' ◦n (x)−t ∈ F(x), where ' ◦n = '◦'◦� � �◦ ' is the n-fold iterate of ', is absolutely irreducible over F; we compute a recursion for its discriminant. Let F' be the field obtained by adjoining to F all roots (in a fixed F) of �n(x, t) for all n ≥ 1; its Galois group Gal(F'/F) is the iterated monodromy group of '. The iterated extension F' is finitely ramified over F if and only if ' is post-critically finite. We show that, moreover, for post-critically finite ', every specialization of F'/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 post-critically finite polynomials. We discuss the wildness of rami- fication in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogene number fields that arise from the construction.
Subfields: Artificial Intelligence, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/54ad637cb857bf608d008ff093429aed1a980454
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Citations in the corpus
- Galois representations from pre-image trees: an arboreal survey (2014, background): “At the extreme of non-generic behavior are the post-critically finite rational functions, which have only a finite set of primes, independent of n, that can ramify in any Kn (this is among the main results in [2] and [6]).”
- The density of prime divisors in the arithmetic dynamics of quadratic polynomials (2006, background): “Since g(fn(γ)) > 0 for infinitely many n, we find, for these n, that |g(fn(γ))/d| is not a square for any d, as desired.”
- Arboreal Galois representations (2007, background): “One can generalize this construction to the setting of iterated polynomial covers of P1K , as first done in (Aitken et al., 2005).”