Isomorphic Rings: \mathbb{F}_5[x]/(x^2+2) and \mathbb{F}_5[x]/(x^2+3)

[JackTheLad]

Homework Statement

Hi guys,

I'm trying to show that $$\mathbb{F}_5[x]/(x^2+2)$$ and $$\mathbb{F}_5[x]/(x^2+3)$$ are isomorphic as rings.

The Attempt at a Solution

As I understand it, I have to find the homomorphism $$\phi:R\to S$$ which is linear and that $$\phi(1)=1$$.

I'm just struggling to find what I need to send $$x$$ to in order to get this work.

 

[JackTheLad]

Actually, I think x --> 2x might do it, because

$$x^2 + 2 \equiv 0$$
$$(2x)^2 + 2 \equiv 0$$
$$4x^2 + 2 \equiv 0$$
$$4(x^2 + 3) \equiv 0$$
$$x^2 + 3 \equiv 0$$

Is that all that's required?