  Given $F$ a real abelian field, $p$ an odd prime and $\chi$ any Dirichlet
character of $F$ we give a method for computing the $\chi$-index $\displaystyle
(H^1(G_S,\mathbb{Z}_p(r))^\chi: C^F(r)^\chi)$ where the Tate twist $r$ is an
odd integer $r\geq 3$, the group $C^F(r)$ is the group of higher circular
units, $G_S$ is the Galois group over $F$ of the maximal $S$ ramified algebraic
extension of $F$, and $S$ is the set of places of $F$ dividing $p$. This
$\chi$-index can now be computed in terms only of elementary arithmetic of
finite fields $\FM_\ell$. Our work generalizes previous results by Kurihara who
used the assumption that the order of $\chi$ divides $p-1$.
