Archimedean algebra of a number field

Core idea

Let $K/\mathbb Q$ be a number field of degree $m=r_1+2r_2$ , with real embeddings $\sigma_1,\ldots,\sigma_{r_1}$ and representatives $\tau_1,\ldots,\tau_{r_2}$ of the complex-conjugate pairs. Its Archimedean algebra is

K_\infty=\mathbb R^{r_1}\times\mathbb C^{r_2}\cong K\otimes_{\mathbb Q}\mathbb R.

The Minkowski embedding is $\iota(a)=(\sigma_1(a),\ldots,\sigma_{r_1}(a),\tau_1(a),\ldots,\tau_{r_2}(a))$ , with coordinatewise algebra operations.

Integral lattice

An integral basis of $\mathcal O_K$ maps to a real basis of $K_\infty$ . Consequently $\iota(\mathcal O_K)$ is a full lattice of real rank $m$ . Some metric conventions rescale complex coordinates by $\sqrt2$ ; that rescaling is not used in this algebra embedding.

Examples

For $K=\mathbb Q$ , this is $\mathbb R$ with lattice $\mathbb Z$ . For $K=\mathbb Q(i)$ , it is $\mathbb C$ with lattice $\mathbb Z[i]$ . For a real quadratic field, it is $\mathbb R^2$ , using both real embeddings.

References

J. S. Milne, Algebraic Number Theory, version 3.08, 2020. Chapter 4, lattices and the embedding used in the finiteness theorem.