$K\otimes_{\mathbb Q}\mathbb R \cong \mathbb R^{r_1}\times\mathbb C^{r_2}$

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted this Lean statement, at the explicit request of the account owner. Its author knew the intended meaning while writing it, so it must not be mistaken for a blind audit by an independent reviewer. Please compare it against the Lean code itself.

Throughout, $K$ is a number field (a field of characteristic zero that is finite-dimensional over $\mathbb Q$), $\mathcal O_K$ is its ring of integers and $\mathcal O_K^\times$ the unit group. $r_1$ denotes the number of real infinite places of $K$ and $r_2$ the number of complex infinite places (each complex place corresponds to a pair of complex-conjugate embeddings $K\to\mathbb C$). Statement. For every number field $K$ there exists an isomorphism of $\mathbb R$-algebras between the tensor product $\mathbb R\otimes_{\mathbb Q}K$ (an $\mathbb R$-algebra via the left factor) and the product ring $\mathbb R^{r_1}\times\mathbb C^{r_2}$, where $\mathbb R^{r_1}$ means functions from a set with $r_1$ elements to $\mathbb R$, $\mathbb C^{r_2}$ functions from a set with $r_2$ elements to $\mathbb C$ (with $\mathbb C$ viewed as an $\mathbb R$-algebra), with componentwise operations. Only existence of such an isomorphism is asserted, not any particular one.