Can anyone explain to me how to show two quotient rings are isomorphic? For my particular case. Both quotient rings are based off ideals in the ring $\mathbb Z_3[X]$: $$ \mathbb Z_3[X]\big/(X^3-X+1)\mathbb Z_3[X] \cong \mathbb Z_3[X]\big/(X^3-X^2+1)\mathbb Z_3[X] $$
There are at least two arguments you can use, special to this particular case. Before I say anything about these, let me say that it’s not clear to me whether, in the general situation, where you have a groundfield $k$ and two irreducible polynomials $f$ and $g$ of the same degree, you an easily determine an isomorphism between $k[x]/(f)$ and $k[x]/(g)$. Indeed, there may be various ad hoc methods of deciding that the two are not isomorphic, but I, being only partially educated here, am unaware of any general such.
Back to your two examples. The polynomials are $f(X)=X^3-X+1$ and $g=X^3-X^2+1$. Ah, but $g(X)=X^3f(1/X)$, telling you immediately how to get an isomorphism. If $\alpha$ is a root of the one and $\beta$ is a root of the other, your isomorphism can send $\alpha\mapsto1/\beta$.
More generally but less explicitly, both are cubic irreducibles over $\mathbb F_3$, so that in both cases the quotient is a field of cardinality $3^3=27$; but the general theory says that there is, up to isomorphism, only one field of each possible finite cardinality $q=p^m$. So the two fields have to be isomorphic. The theory does not construct an isomorphism for you, however, far as I know.
- $X^3-X+1$ and $X^3-X^2+1$ are irreducible polynomials in $\mathbb Z_3[X]$.
- $\mathbb Z_3[X]\big/(X^3-X+1)\mathbb Z_3[X]$ and $\mathbb Z_3[X]\big/(X^3-X^2+1)\mathbb Z_3[X]$ are both finite fields with $27$ elements.
- Finite fields having the same number of elements are isomorphic.
Suppose we wished to show that a quotient ring $A/\mathfrak a$ is isomorphic to another ring $B$. A very useful technique is to define a surjective ring homomorphism $\phi:A\to B$ such that $\ker \phi=\mathfrak a$. That $A/\mathfrak a\simeq B$ then follows from the first isomorphism theorem.