In the proof of Dirichlet's unit theorem, in Algebraic number theory by Samuel, there is a step in the proof that i don't understand. (p.73 in the french edition).
He first introduces the logarithmic embedding $L$ of a number field $K-\{0\}$ in $\mathbb{R}^{r_1+r_2}$ (these are the number of real embeddings and the number of complex embeddings.) Then he proves that the kernel of the restriction of the logarithmic embedding to the algebraic integers $A$ is a finite cyclic group $G$ formed by the roots of $1$ in $K$. Then he says that $L(A)$ is a discrete subgroup of $\mathbb{R}^{r_1+r_2}$, and thus is a free $\mathbb{Z}$-module of rank $s\leq r_1+r_2$.
This is the part i get stuck with : he says that since $L(A^*)$ is free, $A^*$ is isomorphic to $G \times L(A^*) = G\times \mathbb{Z}^s$. Why is $A^*$ is isomorphic to $G \times L(A^*)$ ?
Thanks.
$G$ is the torsion part of $A^*$. By the fundamental theorem of abelian groups, $A^* \cong G \times \mathbb Z^t$ for some $t$. Thus, $L(A^*) \cong A^*/ \ker L = A^*/ G \cong \mathbb Z^t$ and so $ A^* \cong G \times L(A^*)$.