Question

How to prove that a polynomial is irreducible over the integers?

Answer 1

When a polynomial is integer valued, one may appeal to Gauss's lemma which states that if the coefficients of a non-constant polynomial f are relatively prime and f is irreducible in Z[X], then f is irreducible in Q[X].

This allows you to use the structure of Z[X] which is richer than that of Q[X] to prove irreducibility. For example for every prime p there is a reduction map from Z[X] to Fp[X]. If the reduction of f is irreducible over Fp[X] then f is irreducible over Z[X].

Note that for your polynomial, x4−10x2−19mod3 is irreducible in F3[X]. Hence, x4−10x2−19 is irreducible in Z[X].

Answer 2

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