Let $L / K$ be Galois extension of number fields with degree $n$, $U_{K},U_{L}$ be an unit group of an integer ring $\mathcal{O}_K,\mathcal{O}_L$ of $K,L$.
$\sigma_1\dots \sigma_{r_K}$ are real embedding of $K$, $\sigma_{r_K+1}\dots \sigma_{r_K+2s_K}$ are complex embedding of $K$ (where $\sigma_{r_K+j}, \sigma_{r_K+s_K+j}$ are complex conjugate). We set $r\colon=r_K+s_K-1$.
Book now I'm reading says Artin proved following theorem by improving Herbrand's theorem.
There exist $r+1$ units $\eta_1\dots\eta_{r+1}\in U_L$ such that
(a) For all $i=1\dots r+1$, $N_{L/K}\eta_i=1$ or already $N_{F/K}\eta_i=1$ ($F$ is real subfield of $L$)
(b) $U'\colon=\left\langle \{\sigma_i(\eta_j)\}_{i=1\dots n,\ j=1\dots r+1}\cup U_K \right\rangle$ is finite index subroup of $U_L$
But this is not valid statement I think. It seems to $[L:F]=1,2$, but I'm bit confused existence of $F$, when $\eta_i$ belongs to $F$ and so on. So I want ask you to strictly state this in details and prove this.