Isomorphism of quotient ring of polynomial ring

Question

Are $F_3[x]/(x^2-2)$ and $F_3[x]/(x^2-2x-1)$ isomorphic?

I know that $x^2-2$ and $x^2-2x-1$ are irreducible but how to determine if they are isomorphic or not?

Here, $F_3$ means finite field of order 3.

Answer 1

One of the most important facts about finite fields is that all finite fields of the same size are isomorphic!

Finding the isomorphism can be a little trickier. For a problem expressed like yours, the most direct approach would be something like finding a root of the polynomial $t^2 - 2$ in the field $\mathbf{F}_3[x] / (x^2 - 2x - 1)$.

Aside: it sometimes helps to use different indeterminate variables to keep everything straight: e.g. let your two fields be $\mathbf{F}_3[x]/(x^2 - 2)$ and $\mathbf{F}_3[y]/(y^2 - 2y - 1)$.

Answer 2

Hint: Note that $x^2-2x-1 = (x-1)^2 - 2$. Does this suggest an isomorphism?

Answer 3

There's a well known result about finite fields which states that finite fields are completely determined by their cardinality. As you said the two polynomials are irreducible hence the quotient are two $F_3$ vector spaces of dimension $2$ and so they have both cardinality $3^2=9$. By the theorem I've named above it follows that such fields must be isomorphic.