I want to show that $\Bbb R[x] / \langle x^2 + 1 \rangle \simeq \Bbb C$, so note that $x^2+1$ has no zeros in $\mathbb{R}$, therefore the polynomial is irreducible over $\mathbb{R}$, so $\langle x^2+1 \rangle$ is a maximal ideal in $\mathbb{R}[x]$ and $\mathbb{R}[x]/ \langle x^2+1 \rangle$ is a field, in fact, is an extension of $\mathbb{R}$ of degree $2$.
It seems that from this it is possible to establish the isomorphism relation, but I don't see how to continue; any suggestions?
