Given a $\mathbb{Z}$-module $M = \langle 1, \omega_1, \dots, \omega_m \rangle$ contained in the algebraic number field $K = \mathbb{Q}(\omega_1, \dots, \omega_m)$, there is an irreducible form $F(x_0, x_1, \dots, x_m)$ of degree $n$ defined as $F(x_0,x_1, \dots, x_m) = N(x_0 + x_1\omega_1+\dots+x_m\omega_m)$ associated with it, where $N$ denotes the norm $N_{K/\mathbb{Q}}$, $x_0, \dots, x_m$ taking values from $\mathbb{Z}$.
I wonder whether $M$ is unique with respect to the form $F$ in the following sense: suppose $N = \langle \xi_0, \xi_1, \dots, \xi_m \rangle$ is another $\mathbb{Z}$-module in an algebraic number field $K' = \mathbb{Q}(\xi_0, \xi_1, \dots, \xi_m)$, such that $F = N_{K'/\mathbb{Q}}(x_0\xi_0+x_1\xi_1+\dots+x_m\xi_m)$, then $\{\xi_0, \xi_1, \dots, \xi_m\} = \{1\sigma, \omega_1\sigma, \dots, \omega_m\sigma\}$ for some $\sigma \in \operatorname{Gal}(\bar{K}/\mathbb{Q})$, where $\bar{K}$ denotes the normal closure of $K$.
My attempts: I can only prove that the two sets above differ by a multiple of some algebraic number $\alpha$, or equivalently $M\sigma = \alpha N$, by unique factorization of polynomial ring over $\mathbb{C}$. I cannot further prove or disprove it.
The origin of the problem: The problem occurs to me when reading Borevich & Shafarevich's Number Theory, where the author states that different sets of generators of the same module $M$ give rise to equivalent forms.