Let k a number field whose $p$-class group $C_k$ of type $(p,p)$ i.e $C_k \cong \mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}$. Let $k^{(1)}$ be the Hilbert class field of $k$ and $k^{(2)}$ the Hilbert class field of $k^{(1)}$, for a positive integer $n$, we define $k^{(n)}$ is the Hilbert class field of $k^{(n-1)}$...
We define the class fields tower of k as: $k \subset k^{(1)} \subset k^{(2)}\subset k^{(3)} \subset.....\subset k^{(n)}\subset.....$
I need to prove that the class field tower of $k$ terminate in the second step i.e $k^{(2)} = k^{(3)}= k^{(4)}=....$
This is only known to be true for $p = 2$. For odd $p$, Wingberg (On the maximal unramified p-extension of an algebraic number field, J. Reine Angew. Math. 440 (1993), 129–156) claims to have proved that there exist infinite $p$-class field towers for certain fields with class group of type $(p, p)$. I seem to remember that there were problems with some of his claims but at least it shows that the problem is not exactly of the homework type.
For examples in which the towers have length $3$ see https://arxiv.org/abs/1601.00179 .