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Isomorphic rings

  1. Mar 21, 2010 #1
    1. The problem statement, all variables and given/known data
    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.

    3. 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.
  2. jcsd
  3. Mar 21, 2010 #2
    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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