On number fields towers defined by iteration of polynomials

摘要

Let (K-n)(n >= 1) be a tower of number fields whose defining polynomials are iterates of a polynomial f. We show that the sequence of class numbers (h(K-n))(n >= 1) satisfies h(K-n) vertical bar h(Kn+1) when f is a monic Eisenstein polynomial. When f(x) = x(2) - c is a quadratic polynomial, we also determine when the ring of integers O-Kn, equals Z[a(n)], where a(n) is a root of f(n).

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