It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. The question is, can irreducibility be (deterministically) tested any faster? I cannot seem to find anything sensible, but maybe I am not looking in the right places...
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Some hints are perhaps contained in the references here:http://mathoverflow.net/questions/73031/can-irreducibility-of-polynomials-be-figured-out-in-polynomial-time. There are comparisons with testing primality of integers. – Dietrich Burde Nov 20 '13 at 20:25