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?