An Analogue That Fails
The analogous conjecture with the integers replaced by the one-variable polynomial ring over a finite field is false. For example, Swan noted in 1962 (for reasons unrelated to Hypothesis H) that the polynomial
over the ring F2 is irreducible and has no fixed prime polynomial divisor (after all, its values at x = 0 and x = 1 are relatively prime polynomials) but all of its values as x runs over F2 are composite. Similar examples can be found with F2 replaced by any finite field; the obstructions in a proper formulation of Hypothesis H over F, where F is a finite field, are no longer just local but a new global obstruction occurs with no classical parallel, assuming hypothesis H is in fact correct.
Read more about this topic: Schinzel's Hypothesis H
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