$t>0$. In particular, an $\mathcal{O}_K$ lattice is closed under multiplication by elements of $\mathcal{O}_K$. The Euclidean norm in $E$ is induced by the trace form. Notice that $K$ is naturally embedded in $E$. In the cases when $K$ is either totally real or a totally imaginary extension of a real number field (CM-field) some notational simplifications can be made. For instance, we can write the trace form as $$\text{tr}(x\overline{y}) = \sigma_1(x)\overline{\sigma_1(x)} + \ldots + \sigma_n(x)\overline{\sigma_n(x)}.$$

We discuss the average behavior of a general reduction from algebraic number theory [@KosiOngOggier; @Our; @Adaptative], defined in the sequel. A prime $p$ is said to split completely if $p\mathcal{O}_K$ can be factored into the product of prime ideals $\mathfrak{p}_1 \mathfrak{p}_2 \cdots \mathfrak{p}_n$.

Let $p$ be a prime that splits completely, and $\mathfrak{p}$ an ideal above $p$. Consider $\pi: \mathcal{O}_K \to \mathcal{O}_K/\mathfrak{p} \simeq \mathbb{F}_p$ a projection onto $\mathfrak{p}$ and $\sigma$ the canonical embedding. Let $\Lambda=\sigma(\mathcal{O}_K)^t$ (the canonical embedding is applied componentwise). Take $$\phi_p:\Lambda\to\mathbb{F}_p^t$$ $$\phi_p(\sigma(x_1,\ldots,x_t))=(\pi(x_1),\ldots,\pi(x_t))$$ and define $\Lambda_p(\mathcal{C})=\phi_p^{-1}(C) \subset \mathbb{R}^{nt}$. \[def:con2\]

The ensemble induced by Definition \[def:con2\] is non-degenerate.

The minimum algebraic norm of an element of $\mathfrak{p}$ is greater or equal than $p$. Hence $\Lambda_p=\sigma(\mathfrak{p})^t$ has minimum norm at least $\sqrt{n} p^{1/n}$, finishing the proof.

