Principalization of ideals in abelian extensions of number fields
  We give the complete proof of a conjecture of Georges Gras which claims that, for any extension $K/k$ of number fields in which at least one infinite place is totally split, every ideal $I$ of $K$ principalizes in the compositum $Kk^{ab}$ of $K$ with the maximal abelian extension $k^{ab}$ of $k$ 
Sebastien Bosca (IMB), Georges Gras (aucun), Jean-Fran\c{c}ois Jaulent   (IMB)