$\mathbb{F_2}[x]/\langle x^2-2\rangle \cong \mathbb{F_2[x]/\langle x^2-3\rangle }$

Question

Prove that the following rings are isomorphic, where $\mathbb{F_2} = \mathbb{Z}/2\mathbb{Z}$

$$\mathbb{F_2}[x]/\langle x^2-2\rangle \cong \mathbb{F_2[x]/\langle x^2-3\rangle }$$

My attempt:

By checking all elements of the field $\mathbb{F_2}$ we can conclude that both $x^2-2$ and $x^2-3$ are reducibles.

Could you point me towards the direction of the correct proof.

Answer 1

$x^2-2\equiv x^2\pmod {2}$, so it is reducible. Also note $x^2-3\equiv x^2+1\equiv (x+1)^2\pmod{2}$.

Then you can take the map from $F_2[x]\to F_2[x]/(x+1)^2$ that sends $x\mapsto x+1$, and confirm it gives rise to an isomorphism of $F_2[x]/(x^2)$ with $F_2[x]/(x+1)^2$ via the first isomorphism theorem.