Another way to determine if a polynomial is irreducible over a ring, is to check if the zeros of the polynomial are in the ring, ie, does p have a linear factor.

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Another way to determine if a polynomial is irreducible over a ring, is to check if the zeros of the polynomial are in the ring, ie, does p have a linear factor.

For Z_5, write a "for" loop to check if p evaluated at every element of Z_5 is zero.

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If a (monic) polynomial over a (commutative) ring (with one) has a linear factor, than it is of course reducible. But the converse is obviously false, e.g. $(x^2+2)^2$ over integers, even over reals, or as is over some finite field...