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Are $F_2[x]/\langle x^2\rangle$ and $F_2[x]/\langle x^2+x\rangle$ isomorphic? I know they both have $4$ elements: $\{0,1,x,x+1\}$, but how do I define an isomorphism between them?

HSN
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1 Answers1

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Hint:

$\mathbb F_2[x]/\langle x^2\rangle$ has nonzero nilpotent elements.

$\mathbb F_2[x]/\langle x^2+x\rangle \cong \mathbb F_2[x]/\langle x\rangle \times \mathbb F_2[x]/\langle x+1\rangle \cong \mathbb F_2 \times \mathbb F_2$ and so has no nonzero nilpotent elements.

It is instructive to see how their multiplication tables differ.

lhf
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  • See also https://math.stackexchange.com/questions/279388/there-are-at-least-three-mutually-non-isomorphic-rings-with-4-elements – lhf May 27 '20 at 14:23
  • So they are not isomorphic after all, thanks! –  May 27 '20 at 14:25