I am looking at the Eisenstein criterion and this works for unique factorization domain. Now I am asking myself if $\Bbb Q$ is a unique factorization domain? I would yes because it is a field. Now why we use often the criterion to $\Bbb Z$ and conclude with the Gauss lemma that it is irreducible over $\Bbb Q$ instead using it directly to $\Bbb Q$?
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4$\Bbb Q$ is a UFD, but it has no prime elements as it is a field, so the Eisenstein criterion isn't really useful to apply there directly. – leoli1 Feb 03 '21 at 21:16
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1Ah it has no prime element because every element has an inverse (of course except 0), right? – Frederick Manfred Feb 03 '21 at 21:19
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1Yes, that is the reason. – leoli1 Feb 03 '21 at 21:20