I have known that there is an irreducible polynomial for any degree over rational field. However,for other fields such as finite field and extension fields for the rational field,what is the situation for the degree of irreducible polynomials.
Moreover, is there any criterions for this property?

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    $\begingroup$ Over finite fields and number fields (finite extensions of $\mathbb Q$) there are irreducible polynomials of any degree. $\endgroup$ – Wojowu Feb 17 '18 at 9:26
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    $\begingroup$ You have also $\Bbb Q_p$ which has this property. I guess it is sufficient to take the fraction field of a UFD, which has at least one prime element $p$ (to apply Gauss' lemma and Eisenstein's critertion on $X^n - p$). $\endgroup$ – Watson Feb 17 '18 at 9:27
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    $\begingroup$ Any quasi-finite field has a (unique) extension of degree $n$ for any integer $n \geq 1$. $\endgroup$ – Watson Feb 17 '18 at 9:37

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