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This is my homework, I don't have any idea, so I want a hint for this problem.

Let $R_1 = \mathbb{Z}_{11}[X]/\langle X^2 - 2\rangle$ and $R_2 = \mathbb{Z}_{11}[X]/\langle X^2 - 3\rangle$. Show that $R_1\approxeq R_2$.

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The polynomial $X^2-2$ is irreducible over $\mathbb Z_{11}$, so the quotient $\mathbb Z_{11}[X]/(X^2-2)$ is a field. On the other hand, $X^2-3=(X-5)(X-6)$ is a reducible polynomial, so that the quotient $\mathbb Z_{11}[X]/(X^2-3)$ is not a field: indeed, it is not even a domain.

The two quotients are therefore not isomorphic.

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