Isomorphic rings

  • Thread starter JackTheLad
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  • #1
JackTheLad
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Homework Statement


Hi guys,

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

The Attempt at a Solution



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

I'm just struggling to find what I need to send [tex]x[/tex] to in order to get this work.
 

Answers and Replies

  • #2
JackTheLad
7
0
Actually, I think x --> 2x might do it, because

[tex]x^2 + 2 \equiv 0[/tex]
[tex](2x)^2 + 2 \equiv 0[/tex]
[tex]4x^2 + 2 \equiv 0[/tex]
[tex]4(x^2 + 3) \equiv 0[/tex]
[tex]x^2 + 3 \equiv 0[/tex]

Is that all that's required?
 

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