I have a proof that says $\mathbb Q[x]/(x^2+2)$ and $\mathbb Q[x]/(x^2-2)$ are not isomorphic.
However I feel that it is not good one...
First I see that $x^2+2$ and $x^2-2$ are irreducible in $\mathbb Q$. Then I notice that $x^2=2$ and $x^2=-2$.
Now, if we take $(a+bx)(c+dx)$ on both, we end up
$ac-2bd + (ad + bc)x$ and $ac+2bd + (ad + bc)x$.
Can I now say, that these fields do not have the same structure, so they cannot be isomorphic?
I do not want exact answer (homework :) ), but a hint would be nice to guide me into right direction.