On the $p$-rank of class groups of $p$-extensions

We prove a local-global principle for the embedding problems of global fields with restricted ramification. By this local-global principle, for a global field $k$, we use only the local information to give a presentation of the maximal pro-$p$ Galois group of $k$ with restricted ramification, when some Galois cohomological conditions are satisfied. For a Galois $p$-extension $K/k$, we use our presentation result for $k$ to study the structure of pro-$p$ Galois groups of $K$. Then for $k=\mathbb{Q}$ and $k=\mathbb{F}_q(t)$ with $p\nmid q$, we give upper and lower bounds for the rank of $p$-torsion group of the class group of $K$, and these bounds depend only on the structure of the Galois group and the inertia subgroups of $K/k$. Finally, we study the $p$-rank of class groups of cyclic $p$-extensions of $\mathbb{Q}$ and the $2$-rank of class groups of multiquadratic extensions of $\mathbb{Q}$, for a fixed ramification type.