How do I prove that the following rings are not isomorphic to each other? I have already proved that $R \times R$ is not isomorphic to $C$. I also need to show that $C$ is not isomorphic to $R[x]/(x^2)$. Thanks
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Do you mean $\mathbb{R}\times\mathbb{R}$ and $\mathbb{C}$? – Thorgott Mar 02 '20 at 12:51
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@Thorgott yes I do – citizenfour Mar 02 '20 at 13:32
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@DietrichBurde I’m sorry I don’t understand what you mean – citizenfour Mar 02 '20 at 13:33
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I am sorry, let me try again. We have that $\Bbb R[x]/(x^2-a)$ is not isomorphic to $\Bbb R\times \Bbb R$ by the duplicate. Now take $a=0$. This answers your question. Of course $\Bbb R[x]/(x^2)$ has zero divisors, so cannot be a field, so cannot be isomorphic to $\Bbb C$. Is it clearer now? – Dietrich Burde Mar 02 '20 at 13:35
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@DietrichBurde thanks for replying so fast but I’m not sure how the duplicate shows that because the answer uses that $R = \bb{Q}$ – citizenfour Mar 02 '20 at 13:37
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Ah, I see. I should have added this post. – Dietrich Burde Mar 02 '20 at 13:51