Let $p$ be a prime number and $F$ be a number field. Since Iwasawa’s works, the behaviour of the $p$-part of the ideal class group in the ${\mathbb{Z}}_p$-extensions of $F$ has been well understood. Moreover, M. Grandet and J.-F. Jaulent gave a precise result about its abelian $p$-group structure.On the other hand, the ideal class group of a number field may be identified with the torsion part of the $K_0$ of its ring of integers. The even $K$-groups of rings of integers appear as higher versions of the class group. Many authors...