Given a Field of numbers $K:= \mathbb{Q}(\theta)$, $\mathbb{Q}\subset K$ of degree $n$. The Norm of $\alpha \in K $ is defined as $N_K(\alpha) = \prod_{i=1}^{n}\sigma_i(\alpha)$. Where $\sigma_i: K \rightarrow \mathbb{C}$ are the unique monomorphisms (injective Field homomorphisms) such that $\sigma_i(\theta) = \theta_i$, where $\theta_i$ are the roots (in $\mathbb{C}$) of the minimal polynomial of $\theta$.
Now, if i have a $(\omega_j)_{j=1}^n$ a $\mathbb{Q}$-basis of $K$: $N(\alpha=\sum_jc_j\omega_j)= \prod_i\sigma_i(\sum_jc_j \omega_j) = \prod_i\sum_j\sigma_i(c_j\omega_j)=\sum_j c_j\prod_i\sigma_i(\omega_j) = \sum_jc_jN(\omega_j)$
From which we can deduce for $\beta=\sum_jd_j \omega_j$ $N(\alpha + \beta) = \sum_j(c_j + d_j)N(\omega_j) = \sum_jc_jN(\omega_j)+ \sum_jd_jN(\omega_j) = N(\alpha) + N(\beta)$
But this can't be right... Am i going nuts?
Note that $$\prod_i\sum_j\sigma_i(c_j\omega_j)=\sum_\alpha \prod_i \sigma_i(c_{\alpha_i}\omega_{\alpha_i}) \ne \sum_j c_j\prod_i\sigma_i(\omega_j)$$ where the sum over $\alpha=(\alpha_1, ...,\alpha_n)$ exhausts every $n$-tuple such that $1 \le \alpha_i \le n$ for each $i$.