We have seen that $\Omega_{\mathbb{Q}}^{X^2-2}\ncong\Omega_{\mathbb{Q}}^{X^2-3}$ from this answer.
But why $\Omega_{K}^{X^2-2}\cong\Omega_{K}^{X^2-3}$ for $K=\mathbb{F}_5$?
$\Omega$ being the splittingfield of the polynomial over the given field.
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What does this $\Omega$ notation mean? – Qiaochu Yuan Jan 25 '16 at 17:17
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The splittingfield of the polynomial over the given field – Grzegorz Baltissen Jan 25 '16 at 17:18
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1Once you know that a finite field is uniquely determined by its cardinality, it suffices to verify that $x^2 - 2$ and $x^2 - 3$ are either both reducible or both irreducible over $\mathbb{F}5$. It happens that they're both irreducible, so the splitting field is $\mathbb{F}{25}$ in both cases. – Qiaochu Yuan Jan 25 '16 at 17:19
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What if i consider your comment as an answer? – Grzegorz Baltissen Jan 25 '16 at 17:24