The problem:
Determine if $\mathbb Z_2[x]/(x^2+1)$ and $\mathbb Z_2[x]/(x^3+1)$ are isomorphic or not, and justify your answer.
(This is a homework problem.)
I think the rings are not isomorphic. My justification goes:
Elements of $\mathbb Z_2[x]/(x^2 + 1)$ are $\overline{0}$, $\overline{1}$, $\overline{x}$, $\overline{x + 1}$.
Elements of $\mathbb Z_2[x]/(x^3 + 1)$ are $\overline{0}$, $\overline{1}$, $\overline{x}$, $\overline{x + 1}$, $\overline{x^2}$, $\overline{x^2 + 1}$, $\overline{x^2 + x}$, $\overline{x^2 + x + 1}$.
Since the number of elements are different, the rings are not isomorphic.
Is it sufficient to conclude that the rings are not isomorphic just because they contain a different number of elements?
Also, if the problem is changed so that I cannot enumerate the elements like in the justification above (say, if I were to determine if $\mathbb Q[x]/(x^2+1)$ and $\mathbb Q[x]/(x^3 + 1)$ are isomorphic as rings), is there a general strategy that tackles the problem?
Best Answer
Yes, if the cardinalities are different then there is no bijection between the rings, and so no isomorphism.
Over $\mathbb{Q}$ the problem is a bit more interesting. Note that the polynomial $x^2+1$ is irreducible over $\mathbb{Q}$, and so $\mathbb{Q}[x]/(x^2+1)$ is a field. On the other hand, $\mathbb{Q}[x]/(x^3+1)$ is not a field, because $x^3+1$ is reducible over $\mathbb{Q}$.