Inclusion of rings and induced map, fibers? Consider the inclusion of rings $\mathbb{R}[X] \subset \mathbb{C}[X]$ and the corresponding induced map $\phi: \text{Spec}\,\mathbb{C}[x] \to \text{Spec}\,\mathbb{R}[X]$. Can someone give me an explicit algebraic and geometric description of all the fibers?
 A: The prime ideals of $\mathbb{R}[x]$ have the form $\langle f\rangle$, where $f$ is one of:


*

*$f(x) = 0$;

*$f(x) = x - \lambda$, where $\lambda \in \mathbb{R}$;

*$f(x) = x^2 + bx + c$, where $b, c \in \mathbb{R}$ and $b^2 - 4c < 0$.


In the first case, the fiber over $\langle0\rangle \subseteq \mathbb{R}[x]$ is $\langle 0\rangle \subseteq \mathbb{C}[x]$, and corresponds algebraically to the affine scheme associated with the ring$$\mathbb{R}(x) \otimes_{\mathbb{R}[x]} \mathbb{C}[x] = \mathbb{C}(x).$$In the second case, the fiber over $\langle x - \lambda \rangle$ corresponds algebraically to the affine scheme associated with the ring$$\mathbb{R}[x]/\langle x - \lambda \rangle \otimes_{\mathbb{R}[x]} \mathbb{C}[x] = \mathbb{C}[x]/\langle x- \lambda\rangle.$$Geometrically, this means that the fiber over the "point" $\langle x - \lambda\rangle$ is the single point $\lambda \in \mathbb{A}_\mathbb{C}^1$.
In the third case, we can write $f(x) = x^2 + bx + c = (x - \alpha)(x - \overline{\alpha})$ where $\alpha \in \mathbb{C}\setminus \mathbb{R}$. Algebraically, the fiber over $\langle x^2 + bx + c\rangle$ is the affine scheme corresponding to the ring$$\mathbb{R}[x]/\langle x^2 + bx + c\rangle \otimes_{\mathbb{R}[x]} \mathbb{C}[x] = \mathbb{C}[x]/\langle(x-\alpha)(x-\overline{\alpha})\rangle.$$Geometrically, the fiber over the "point" $\langle x^2 + bx + c\rangle$ is the two-point set $\{\alpha, \overline{\alpha}\} \subseteq \mathbb{A}_\mathbb{C}^1$.
