How do I show that $\mathbb{Q}[X]/(X^2 + X + 1)$ and $\mathbb{Q}[X]/(X^2 + 1)$ are not isomorphic?
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2For the second part, see $\mathbb Q[x]/(x^2+1)$ is not isomorphic to $\mathbb Q[x]/(x^2+2)$, and apply it in the same way to $x^2+x+1$. Derek Holt's comment also answers this immediately. – Dietrich Burde Jun 01 '16 at 13:37
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Now it is a duplicate of the second link, after you have edited the question. Since there is an element $a$ with $a^2+1=0$ in the second field, it must be also an element $b$ with $b^2+1=0$ in the first field, which is not the case. Hence the two fields are not isomorphic. – Dietrich Burde Jun 01 '16 at 14:29