Let \(K\) be a number field. Denote \(K_{\mathbb{R}} = K \otimes_{\mathbb{Q}} \mathbb{R}\). Then the ring of integers of \(\mathcal{O}_{K}\) of \(K\) forms a discrete subgroup of largest possible rank within \(K_{\mathbb{R}}\).

If \(K\) has \(r_1\) real embeddings and \(r_2\) complex embeddings then after choosing one representative from each pair of complex embeddings, we have the following isomorphism as \(\mathbb{R}\)-algebras. \[\begin{align} K_{\mathbb{R}} \simeq & \mathbb{R}^{r_1} \oplus \mathbb{C}^{r_2}\\ x \otimes 1 \mapsto & (\sigma(x))_{\sigma:K \rightarrow \mathbb{R}\text{ or }\mathbb{C}} \end{align}\]

Uses

Sphere packing

Used in [1] and [2].

See Siegel Mean Value theorem and sphere packing constant.

Also see ideal_lattices.

Dirichlet’s unit theorem

See the page.

Discriminant of a number field

On \(K_{\mathbb{R}}\), we can set the measure natural Lebesgue measure induced by the identification given above. In terms of this measure, \[\sqrt{D_{K}} = \mathop{\mathrm{vol}}(K_{\mathbb{R}} / \mathcal{O}_{K}),\] which is to say that the discriminant of the number field is the square of the volume of the ring of integers with respect to the Minkowski’s embedding.

CAUTION: Here \(K_{\mathbb{R}}\) is embedded with the Minkowski metric, as mentioned in [3]. This differs from the Lebesgue measure ( obtained by identifying \(K_{\mathbb{R}} \simeq \mathbb{R}^{[K:\mathbb{Q}]}\) ) by a factor of \(2^{r_2}\).

References