Are fields $Q[x]/(x^2 -5)$ and $Q[x]/(x^2 + 5)$ isomorphic to each other?

Question

I am unable to solve this. My approach was to prove $Q[x]/(x^2 + 5)$ is isomorphic to $Q[k]$ for some k and then similarly show it for $Q[x]/(x^2 - 5)$ implying there are both isomorphic to each other.

Answer 1

Hint: One is isomorphic to $\mathbb Q[\sqrt 5]$ whereas the other is isomorphic to $\mathbb Q[\sqrt{-5}]$.

Answer 2

They are not isomorphic. One field contains an element $\alpha^2 = -5$ the other does not.

Answer 3

Hints.

Set $\alpha=x+ (x^2-5)\mathbb{Q}[x]$ and $\beta=x+ (x^2+5)\mathbb{Q}[x]$.

• Justify (if you feel it necessary) that any element of $\mathbb{Q}[x]/(x^2-5)$ has the form $a+b\alpha, a,b\in\mathbb{Q}$ (with the usual abuse of notation), while any element of $\mathbb{Q}[x]/(x^2+5)$ has the form $a+b\beta, a,b\in\mathbb{Q}$ . This will be handy for computations.

• Let $f:\mathbb{Q}[x]/(x^2-5)\to \mathbb{Q}[x]/(x^2+5)$ be an isomorphism.

Show that $f(\alpha)^2=5$ (again with the usual abuse of notation). Then prove that $\mathbb{Q}[x]/(x^2+5)$ does not contain any element whose square is $5$.