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Show that $$Q[x]/(x^2-2)$$ is not isomorphic to $$Q[x]/(x^2-3)$$

I was practicing some exercises in Hungerford's abstract algebra book, I could make them all but this one. I think I have to verify if it is irreducible and thus be able to say if it is isomorphic, but I do not know how. I hope you can help me understand.

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  • $\begingroup$ Isomorphic as a ring or as a $\mathbb{Q}$-algebra? $\endgroup$ – Leon Lang Sep 14 '17 at 1:27
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    $\begingroup$ You do understand that the first ring has a square root of $2$, and the second has a square root of $3$, right? Can you show that the first doesn’t have a square root of $3$? $\endgroup$ – Lubin Sep 14 '17 at 1:37
  • $\begingroup$ I believe that Q (rationals) $\endgroup$ – Hector Javier Torres Sep 14 '17 at 1:38
  • $\begingroup$ I added the "field-theory" and the "extension-field" tags to your post. $\endgroup$ – Robert Lewis Sep 14 '17 at 2:50