Relative extensions of number fields and Greenberg’s Generalised Conjecture
Let $p$ be a fixed prime. In this article, we will prove several results concerning Greenberg’s Generalised Conjecture (GGC). On the one hand, we will prove that whenever a slightly stronger form of (GGC) holds for a number field $K$ (which will be the case in most of the examples), then the conjecture also holds for every finite normal $p$-ramified $p$-extension of $K$. On the other hand, we will directly prove that (GGC) holds for certain number fields containing exactly one prime above $p$. These results are based on the insight that the validity of (GGC) for some number field $K$ can be checked by studying $\mathbb Z_p$- and $\mathbb Z_p^2$-extensions of $K$. We will also provide new examples in which (GGC) holds in a non-trivial way.