- How can I show that $\mathbb{Q}[x]/(x^2+2)$ and $\mathbb{Q}[x]/(x^2-2)$ are not isomorphic?
If $\alpha$ and $\beta$ are zeros of $x^2+2$ and $x^2-2$ in certain extensions respectively, we have that $\mathbb{Q}[x]/(x^2+2)\cong\mathbb{Q}(\alpha)$ and $\mathbb{Q}[x]/(x^2-2)\cong\mathbb{Q}(\beta)$.
First I thought that if they were isomorphic then there would be a $\mathbb{Q}$-isomophism mapping $\alpha$ to $\beta$ (for some natural reason that I was missing) and then it would be easy to prove that such $\mathbb{Q}$-homomorphism is not possible, but the truth is that this is not always true.
For example $\mathbb{R}[x]/(x^2+1)$ and $\mathbb{R}[x]/(x^2+x+1)$ are isomorphic to $\mathbb{C}$ and there is no $\mathbb{R}$-homomorphism mapping a zero of $x^2+1$ into a zero of $x^2+x+1$. So I run out of ideas.
- How can I show that $\mathbb{R}[x]/(x^2+x+1)$ is isomorphic to $\mathbb{C}$?