Iterated towers of number fields by a quadratic map defined over the Gaussian rationals

Abstract

An iterated tower of number fields is constructed by adding preimages of a base point by iterations of a rational map. A certain basic quadratic rational map defined over the Gaussian number field yields such a tower of which any two steps are relative bicyclic biquadratic extensions. Regarding such towers as analogues of $\mathbf{Z}_{2}$-extensions, we examine the parity of 2-ideal class numbers along the towers with some examples.