Schinzel's Hypothesis H - An Analogue That Fails

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.