I would like to prove the following:

Let $K_1$, (resp. $K_2$) be Galoisian extension of $\mathbb Q$ of degree $[K_1:\mathbb Q]=n_1$ with ring of integers $\mathcal O_{K_1}$ and Galois group $G_1$ (resp. $\mathcal O_{K_2}$ with degree $n_2=[K_2: \mathbb Q]$ and Galois group $G_2$). One assumes that $GCD(n_1,n_2)=1$. Denote by $L$ the compositum of $K_1$ and $K_2$ and its ring of integers $\mathcal O_L$. Then, there exists an isomorphism.

$H^1(G_1,\mathcal O_{K_1}^{\times})/(\oplus_{p\text{ prime}}\mathbb Z/e_p\mathbb Z)\times H^1(G_2,\mathcal O_{K_2}^{\times})/(\oplus_{p\text{ prime}}\mathbb Z/e_p\mathbb Z)\simeq H^1(G_1\times G_2,\mathcal O_{L}^{\times})/(\bigoplus_{p\text{ prime}}\mathbb Z/e_p\mathbb Z)$

where $e_p$ denotes the ramification index of the prime $p$ in the corresponding extension.

Any help would be welcome.

Thanks in advance

I would like to prove the following:

Let $K_1$, (resp. $K_2$) be a finite Galoisian extension of $\mathbb Q$ of degree $[K_1:\mathbb Q]=n_1$ with ring of integers $\mathcal O_{K_1}$ and Galois group $G_1$ (resp. $\mathcal O_{K_2}$ with degree $n_2=[K_2: \mathbb Q]$ and Galois group $G_2$). One assumes that $GCD(n_1,n_2)=1$. Denote by $L$ the compositum of $K_1$ and $K_2$ and its ring of integers $\mathcal O_L$. Then, there exists an isomorphism.

$H^1(G_1,\mathcal O_{K_1}^{\times})/(\oplus_{p\text{ prime}}\mathbb Z/e_p\mathbb Z)\times H^1(G_2,\mathcal O_{K_2}^{\times})/(\oplus_{p\text{ prime}}\mathbb Z/e_p\mathbb Z)\simeq H^1(G_1\times G_2,\mathcal O_{L}^{\times})/(\bigoplus_{p\text{ prime}}\mathbb Z/e_p\mathbb Z)$

where $e_p$ denotes the ramification index of the prime $p$ in the corresponding extension.

Any help would be welcome.

Thanks in advance