Let $k$ be a number field, which we think as a subset of $\mathbb{C}$, and let $|\!\cdot\!|$ be the absolute value of $\mathbb{C}$. Further, write $N_k$ for the norm of $k$ over $\mathbb{Q}$, and put $n = [k : \mathbb{Q}]$.
Is it true that $|N_k(a + b)|^{1/n} \leq |N_k(a)|^{1/n} + |N_k(b)|^{1/n}$ for all $a,b \in k$ ?
If so, where can I find a reference or a short proof for this fact?
I believe that inequality should be true because I remember that $|N_k(\cdot)|^{1/n}$ is used to extend an absolute value on $\mathbb{Q}$ to $k$, or something similar...
Thanks.