Index for subgroups of the group of units in number fields

We define a sequence of rational integers ${\displaystyle u_i\left(E\right)}$ for each finite index subgroup E of the group of units in some finite Galois number fields K in which prime p ramifies. For two subgroups E' ⊂ E of finite index in the group of units of K we prove the formula ${\displaystyle v_p\left([E:E\prime ]\right)={\sum }_{i=1}^r\{u_i(E\prime )-u_i\left(E\right)\}}$. This is a generalization of results of P. Dénes [3], [4] and F. Kurihara [5].