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Irreducibility

Polynomial factorization (hence also irreducibility testing) over $\mathbb{F}_q[x]$ can be tested done efficiently using the Berlekamp or similar algorithms . For (for polynomial rings over other fields the situation is differentdifferent). See, e.g., herethis PDF for an elementary discussion.
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Irreducibility over $\mathbb{F}_q[x]$ can be tested efficiently using the Berlekamp or similar algorithms. For polynomial rings over other fields the situation is different. See, e.g., here for an elementary discussion.