The simplest non-trivial example of $\mathbb{Z}_p$-extensions happen over imaginary quadratic fields and I've seen a good amount of research done here (from my understanding still a lot to learn). I would think that the next step up in complexity would be looking at the $\mathbb{Z}_p$-extensions of bi-quadratic fields, say $k = \mathbb{Q}(\sqrt{-m}, \sqrt{n})$ where $m \in \mathbb{Z}^+, n \in \mathbb{Z}$ are square free. I've been searching around but can't seem to find any work done in this area. In particular, I'm interested in Iwasawa's $\lambda$-invariant for the cyclotomic $\mathbb{Z}_p$-extension over $k$. Does anyone know of any papers or research that have to do with this? Thanks for any help.
Edit: After a bit of thought I realized that this may not be so interesting. I think that we can determine $\lambda^-$ (the minus part of Iwasawa's lambda invariant for the cyclotomic $\mathbb{Z}_p$ extension) from the $\lambda$-invariants of the two imaginary subfields of $k$. That is, $\lambda^- = \lambda_{\chi_1} + \lambda_{\chi_2}$, where $\chi_1, \chi_2$ are the odd characters belonging the imaginary quadratic subfields. We have $\lambda = \lambda^- + \lambda^+$, but $\lambda^+$ is expected to be 0 by Vandiver's conjecture.
