Do the following ring homomorphisms with identity exist from $R$ to $S$?
a) $R=\mathbb{Q}[X]$, $S=\mathbb{Q}$, $f(X^2-2)=0$
b) $R=\mathbb{R}[X]$, $S=\mathbb{C}$, $f(X^2+4)=0$
I know the following fact:
Let $R$ be a commutative ring with identity, $f \in R[X]$ and $r \in R$, then
$$ \text{ev}_r : R[X] \to R, \text{ ev}_r(f) = f(r)\text{ is a ring homomorphism.}$$
To check b), I use this fact:
We have $2i \in \mathbb{C}$, and $\text{ev}_{2i}(f) = f(2i) = 0$, it holds, also $\text{ev}_{2i}(1_{\mathbb{R}[X]})=1_{\mathbb{C}}.$
How to deal with a)?
I can't use that trick because we have $\sqrt{2} \notin \mathbb{Q}$. And from the definition, it's not clear how to check whether it is a homomorphism.
Hint (for a)): suppose $f$ is a ring homomorphism. What is the value of $f(X)$?