I am trying to show the following: $\Omega^{X^2-2}_\mathbb{Q} \ncong \Omega^{X^2-3}_{\mathbb{Q}}$, but $\Omega^{X^2-\overline{2}}_K \simeq \Omega^{X^2-\overline{3}}_K$ for $K = \mathbb{F}_5$.
Fort the first part, I have the following: $X^2-2 = (X-\sqrt{2})(X+\sqrt{2})$, and $X^2-3 = (X-\sqrt{3})(X+\sqrt{3})$, so $\mathbb{Q}(\sqrt{2}) \neq \mathbb{Q}(\sqrt{3})$.
For the second part, $X^2-\overline{2}$ and $X^2-\overline{3}$ are both irreducible in $K$, since they have no roots. That means that the following are their splitting fields: $$ \mathbb{F}_5[X]/(X^2-\overline{2}), \mathbb{F}_5[X]/(X^2-\overline{3}), $$ respectively. So I am trying to show that these are isomorphic, but how? Am I on the right track so far? Also, can someone explicitly point out how these two polynomials can be written in linear factors in the above two splitting fields?