Small nonzero element in a number-field ideal

Given: A number field $K$ of degree $n$ and signature $(r_1,r_2)$, so $n=r_1+2r_2$, with ring of integers ${\mathcal{O}}_K$ and nonzero integral ideal $\mathfrak{a}\subseteq {\mathcal{O}}_K$ of absolute norm $N\mathfrak{a}$ (Unscaled Minkowski embedding).

Minkowski convex-body theorem at equality: under the Axiom of Choice, if $C\subseteq {\mathbb{R}}^n$ is compact, convex and centrally symmetric and $\mathrm{vol}(C)\ge 2^n\mathrm{covol}(\mathrm{\Lambda })$ for a full lattice $\mathrm{\Lambda }$, then $C$ contains a nonzero point of $\mathrm{\Lambda }$ (Minkowski convex-body theorem at equality, Full Euclidean lattice and covolume).

For $t>0$ the set $X_t=\{(x,z)\in {\mathbb{R}}^{r_1}\times {\mathbb{C}}^{r_2}:{\sum }_i\mathrm{\mid }{x}_i\mathrm{\mid }+2{\sum }_j\mathrm{\mid }{z}_j\mathrm{\mid }\le t\}$ is compact, convex and centrally symmetric, has $\mathrm{vol}(X_t)=2^{r_1}(\pi \mathrm{/}2{)}^{r_2}{t}^n\mathrm{/}n!$, and every point of $X_t$ satisfies ${\prod }_i\mathrm{\mid }{x}_i\mathrm{\mid }{\prod }_j\mathrm{\mid }{z}_j{\mathrm{\mid }}^2\le (t\mathrm{/}n{)}^n$ (Archimedean product region, volume and norm bound).

For a nonzero integral ideal $\mathfrak{a}$ the image $\sigma (\mathfrak{a})$ under the unscaled Minkowski embedding is a full lattice in ${\mathbb{R}}^n$ with $\mathrm{covol}(\sigma (\mathfrak{a}))=2^{-r_2}\sqrt{\mathrm{\mid }{d}_K\mathrm{\mid }}\cdot N\mathfrak{a}$ (Number-field integer rings and ideals are full lattices, Covolume of an integral ideal lattice).

For $0\mathrm{\ne }\alpha \in K$ the norm is $N_{K\mathrm{/}\mathbb{Q}}(\alpha )={\prod }_{i=1}^{r_1}{\sigma }_i(\alpha ){\prod }_{j=1}^{r_2}{\tau }_j(\alpha ){\bar{\tau }}_j(\alpha )$, so $\mathrm{\mid }{N}_{K\mathrm{/}\mathbb{Q}}(\alpha )\mathrm{\mid }={\prod }_i\mathrm{\mid }{\sigma }_i(\alpha )\mathrm{\mid }{\prod }_j\mathrm{\mid }{\tau }_j(\alpha ){\mathrm{\mid }}^2$ (Norm and trace from embeddings, with the inseparable exponent in the norm formula).