Ideal lattices

Let \(K \rightarrow K_{\mathbb{R}}\) be the Minkowski embedding. Let \(\mathcal{I} \subseteq K\) be a fractional ideal. Then, we get that the embedding of \(\mathcal{I}\) is a lattice in \(K_{\mathbb{R}}\).

We can also say that the covolume of this lattice is \(\sqrt{| D_{K}|}\) and that the shortest vector is bounded by

\[\begin{align} \lambda_1(\mathcal{I}) \ge \sqrt{[K:\mathbb{Q}]} \mathop{\mathrm{N}}(I)^{\frac{1}{[K:\mathbb{Q}]}}. \end{align}\]

This follows from the AM-GM inequality. Also knowns as the norm-trace inequality in this context.