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 L_j = L₀(zeta_22+j) so that the fields L_j are the cyclotomic Z₂-extension of L₀. We determine when fourth roots of certain elements of K₀ generate unramified extensions of L_j. 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 L_j(gamma1/4)/L_j 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 L_j/K_j 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.