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There are irreducible monic polynomials over $\mathbb{Z}$ that are reducible modulo every prime number $p$ (e.g. $x^4+1$). Given a finite non-empty set $S$ of primes is there a monic polynomial over $\mathbb{Z}$ that is irreducible modulo the primes in $S$ and is reducible modulo all other primes?

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Perhaps surprisingly, the answer is no for every nonempty $S$.

If $f(x)$ is a degree-$d$ polynomial with integer coefficients, then its factorization modulo a prime $p$ is related to a conjugacy class of the Galois group of $f$ over $\Bbb Q$ (the Frobenius class). In particular, $f$ being irreducible when reduced modulo $p$ is equivalent to the Frobenius class corresponding to a cycle that permutes the $d$ roots of $f$.

In any case, the Chebotarev density theorem says that the set of primes whose Frobenius lies in a particular conjugacy class of its Galois group is either empty or has positive relative density inside the primes; in particular, it cannot be finite and nonempty.

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  • $\begingroup$ what is the definition of the Frobenius conjugacy class at ramified primes? $\endgroup$
    – user700841
    Feb 5, 2020 at 3:58
  • $\begingroup$ Good question! It's not defined—but ramified primes correspond to primes modulo which the polynomial has repeated roots, so in particular is reducible. $\endgroup$ Feb 5, 2020 at 5:34

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