I often encounter trouble in proving isomorphism of things in terms of quotients, say is $\mathbb{Q}$[x]/$(x^5-3)$ is isomorphic to $\mathbb{Q}$[x]/$(x^5-2)$ or $\mathbb{Q}$[x]/$(x^3-3)$? Is there any way to attack such problem? Thank you very much.
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$\mathbb{Q}$[x]/$(x^5-3)$ is not isomorphic to $\mathbb{Q}$[x]/$(x^5-2)$ because $\mathbb{Q}$[x]/$(x^5-3)$ has an element $u$ such that $u^5=3$ but $\mathbb{Q}$[x]/$(x^5-2)$ doesn't. (I'm guessing here, but for these examples it seems a safe guess.)
$\mathbb{Q}$[x]/$(x^5-3)$ is not isomorphic to $\mathbb{Q}$[x]/$(x^3-3)$ because $\mathbb{Q}$[x]/$(x^5-3)$ has dimension $5$ over $\mathbb{Q}$ whereas $\mathbb{Q}$[x]/$(x^3-3)$ has dimension $3$.
These arguments above are typical for this task.
More generally, to prove that $A$ is not isomorphic to $B$ it suffices to find a property of $A$ that does not hold for $B$. The property needs to be invariant under isomorphisms. This happens in other areas as well. For instance, a connected space cannot be homeomorphic to a space that is not connected.
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