A real conic acquires complex points

Given: The objects, hypotheses and conventions in the statement above.

For every field $K$ and scheme $X$, morphisms $\mathrm{Spec}K\to X$ correspond bijectively to pairs $(x,\iota )$ with $x\in X$ and a field embedding $\iota :\kappa (x)\to K$. The identity embedding gives a canonical morphism $\mathrm{Spec}\kappa (x)\to X$, compatible with all scheme morphisms. More generally, for a nonzero local ring $(R,\mathfrak{m})$, morphisms $\mathrm{Spec}R\to X$ correspond to pairs $(x,\phi )$ with a local homomorphism $\phi :{\mathcal{O}}_{X,x}\to R$. Assuming Choice, two field-valued points have the same image in $X$ if and only if they are dominated by a common field-valued point, by compatible embeddings of their fields into a third field. (Field-valued points and local-ring points)

For a field extension $K\mathrm{/}k$ and a $k$-scheme $X$, the inverse image of every affine open $U=\mathrm{Spec}A$ in $X_K$ is $\mathrm{Spec}(A{\otimes }_{k}K)$. These affine charts cover $X_K$ and are compatible on overlaps and with coefficient localizations. (Affine charts after extension of the ground field)

Let $A\to C$ be a unital ring map. For any set of variables $(t_i)$ and any ideal $I\subseteq A[t_i]$, $(A[t_i]\mathrm{/}I){\otimes }_{A}C\cong C[t_i]\mathrm{/}IC[t_i]\mathrm{.}$ Here the extended ideal is generated by the coefficient images of all elements of $I$. For a multiplicative subset $M\subseteq A$, $(M^{-1}A){\otimes }_{A}C\cong {\bar{M}}^{-1}C\mathrm{.}$ These are ring isomorphisms; no flatness, finite-generation or nonzero-ring hypothesis is required. (Presentations and localization under base extension)