Let $K$ be a number field of degree $n$ and let $H$ be a finitely generated subgroup of $\mathcal{O}_K^+$ of rank $n$. Then $H$ has a $\mathbb{Z}$-basis $\omega_1,...,\omega_n $.
$\Delta (\omega_1,...,\omega_n)=\text{Det} (c_{ij})_{n \times n} $ where $c_{ij}=\text{Trace}_{K/\mathbb{Q} }(\omega_i \omega_j) . $
This is the part I don’t understand next.
It says that $H \subset \mathcal{O}_K $ so each $c_{ij} $ is an integer so $\Delta (\omega_1,...,\omega_n )$ is an integer.
My question is why does the fact that $H \subset \mathcal{O}_K$ mean that $\text{Trace}_{K/\mathbb{Q} }(\omega_i \omega_j)$ is an integer? I could understand this if $H$ was a subring of $\mathcal{O}_K$ because then $\omega _i \omega_j $ would be an element of $H$ and then we could use the fact that $\omega_1,...,\omega_n $ is a $\mathbb{Z}$-basis. But $H$ is only an additive subgroup of $\mathcal{O}_K^+$. What am I missing here?