I am reading about the class field tower problem in Cassels-Frohilch. It seems to me that the Hilbert $H(K)$ class field is the smallest Galois extension of $K$ such that every ideal of $K$ becomes principal.
I tried to find a source for this claim (which I hope is true), but I didn't succeed at Neukirch or Milne. I would appetite any answer which explain the situation.
This is not true. Take e.g. $K = {\mathbb Q}(\sqrt{3 \cdot 7 \cdot 11})$; here $Cl(K) \simeq (2,2)$, and every ideal class capitulates in every unramified quadratic extension. This happens always if the class field tower of a number field with noncyclic $2$-class group is abelian. This phenomenon is treated at length in the books on algebraic number theory and class fields by Harvey Cohn.