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

[JackTheLad]

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.

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.

 

[Hurkyl]

Well, what property must the image of x satisfy?


If all else fails, there aren't many possibilities, you could just try them all.

 

[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?

 

[mrbohn1]

Do you have to provide an explicit isomorphism? If not you can just use the fact that finite fields with the same cardinality are isomorphic...both of these fields are generated by adjoining a root of an irreducible quadratic to a field of order 5, and hence both have 25 elements.

 

[JackTheLad]

Yeah, unfortunately I do have to show the explicit isomorphism (we're supposed to do it 'the long way')