Irreducible polynomials which are locally reducible everywhere

Guralnick, Robert; Schacher, Murray; Sonn, Jack

doi:10.1090/S0002-9939-05-07855-X

Irreducible polynomials which are locally reducible everywhere
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by Robert Guralnick, Murray M. Schacher and Jack Sonn PDF

Proc. Amer. Math. Soc. 133 (2005), 3171-3177 Request permission

Abstract:

For any positive integer $n$, there exist polynomials $f(x)\in \mathbb {Z}[x]$ of degree $n$ which are irreducible over $\mathbb {Q}$ and reducible over $\mathbb {Q}_{p}$ for all primes $p$ if and only if $n$ is composite. In fact, this result holds over arbitrary global fields.

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