Unramified Extensions of the Cyclotomic Z_2-Extension of Q(sqrt(d),i)

Abstract

Let F0 = Q((-d)½), K0 = Q(d½), and L0 = Q(d½, i) with d a square-free positive integer such that 2 does not divide d. Let Lj = L0(zeta22+j) so that the fields Lj are the cyclotomic Z2-extension of L0. We determine when fourth roots of certain elements of K0 generate unramified extensions of Lj. In particular, for elements of K0 that are relatively prime to 2 and are generators of principal ideals that are fourth powers, we give explicit congruence conditions under which the fourth root of the element gives an unramified extension. For any such element gamma, we show that if there is some j such that Lj(gamma1/4)/Lj is unramified, then L2(gamma1/4)/L2 is unramified. We also show that when (2) is split in F0, L2(gamma1/4)/L2 is unramified for any such gamma. This result is analogous to a result by Hubbard and Washington in which they work with the cyclotomic Z3-extension of Q((-d)½, zeta3) when 3 does not divide d and consider extensions generated by cube roots of elements in Q((3d)½). However, many more technical problems arise in the present work because the degree of the extension Lj/Kj is not relatively prime to the degrees of the extensions being generated. In order to prove our main results, we also give a congruence condition, which, for any number field K containing i and for any element gamma in K with gamma relatively prime to 2 and gamma a generator of an ideal that is a fourth power, dictates whether or not adjoining a fourth root of gamma to K gives an unramified extension.