Let $K$ be a number field, which we may suppose satisfies $n = [K : \mathbb{Q}] \geq 3$. Let $\mathcal{O}_K$ be the ring of integers of $K$, and let $\{\omega_1, \cdots, \omega_{n}\}$ be a basis of $\mathcal{O}_K$. Define the norm form of $K$ to be the homogeneous polynomial defined by

$\displaystyle N_{K/\mathbb{Q}}(\mathbf{x}) = \prod_{\sigma : K \hookrightarrow \mathbb{C}} \left(\sigma(\omega_1) x_1 + \cdots + \sigma(\omega_n) x_n\right),$

where the product runs over embeddings $\sigma: K \hookrightarrow \mathbb{C}$.

We can extend this definition of norm form to any order $\mathcal{O}$ contained in $\mathcal{O}_K$, by replacing the basis $\{\omega_1, \cdots, \omega_n\}$ with a basis for $\mathcal{O}$.

Now suppose that $\mathcal{O}$ is an order, with basis $\{\nu_1, \cdots, \nu_n\}$, say, and let $N_{\mathcal{O}}(\mathbf{x})$ be the corresponding norm form. Let us take an element $A \in \text{GL}_n(\mathbb{Z})$, and consider $G : = N_{\mathcal{O}}(A \mathbf{x})$. Now restrict $N_{\mathcal{O}}, G$ to the 2-dimensional space defined by $x_3 = \cdots = x_n = 0$, to obtain binary $n$-ic forms $f(x_1, x_2),g(x_1, x_2)$ say.

What can we say about $f,g$ in general? In particular, are their discriminants connected in any way? What about the rings/ideal classes they represent?


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