So I am not sure if the answer to this question is a trivial one. It would be very nice if it were totally true.

Let $K$ be a number field over $\mathbb{Q}$ and let $N_{K/\mathbb{Q}}$ be the norm mapping.

Let us make a small definition for simplicity:

For an ideal $I \subset \mathbb{O}_K$ let $S_I = \{N_{K/\mathbb{Q}}(x) : x \in I\} \subset \mathbb{Z}$

Consider two ideals $\alpha, \beta \subset \mathbb{O}_K$

Is it then always the case that for any $\sigma \in Gal(K/\mathbb{Q})$ that: $S_{\alpha \cdot \beta} = S_{\alpha \cdot \sigma(\beta)}$ ?