Let $\Bbb R[x]$ denote the ring of all polynomials with coefficients in $\Bbb R$. Find a surjective ring homomorphism $\phi : \Bbb R[x] \to \Bbb C$ and compute its kernel.
I don't know how to begin because polynomial can have infinite coefficients but complex numbers are in the form $a+bi$.
Best Answer
The easiest surjective ring homomorphism $\Bbb R[x]\to \Bbb C$ takes the polynomial $x$ to $i\in\Bbb C$, but in fact, assigning any nonreal complex number would work.
Observe that the image of $x$ solely already determines the map as it supposed to be a homomorphism.