Schinzel nd reducibility polynomials covering systems congruences

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Schinzel, A., Reducibility of polynomials and covering systems of congruences. Acta Arith. 13 (1967), 91-101. The scan carries only the digitizer's icm mark with a copyright sign and prints no license line; the publisher's record labels the PDF download "Pobierz zgodnie z CC-BY", which the English site renders "Free download under CC-BY license", naming no version or license URL (https://www.impan.pl/get/doi/10.4064/aa-13-1-91-101, read 2026-10-02).

Motivated by a question of Turan on approximating an integer polynomial by an irreducible one, Schinzel studies when x^n + f(x) is irreducible over the rationals for infinitely many n, and links this to covering systems of congruences a_i mod m_i. Theorem 1 proves the equivalence of two propositions: (A) for every integer polynomial f with f(0) != 0, f(1) != -1 and f not identically 1 there is an arithmetic progression N with x^v + f(x) irreducible for every v in N; and (B) in every finite covering system with m_i > 1 at least one of the quotients m_j/m_i equals q^a for a prime q and an a >= 0, with a_j != a_i mod m_i and either q > 2 or m_i = 1 mod 2 or a_j != a_i mod (m_i/2). By Theorem 2, C implies B and B implies D, where (C) says every finite covering system with m_i > 1 has two equal moduli or an even modulus, and (D) says every such system has at least one modulus dividing another. The technique is polynomial-congruence machinery over Z[x]: Lemma 1 constructs f congruent to prescribed a_i(x) modulo pairwise coprime monic F_i(x) with degree below the product, and Lemma 2 shows for prime q and m <= n that the cyclotomic polynomials X_m and X_n are relatively prime mod q unless n/m = q^a (a >= 0), in which case X_n = X_m^{phi(n)/phi(m)} mod q. Here monic means leading coefficient +1 or -1. Schinzel singles out the consequence that Selfridge's conjecture, that no covering system has distinct odd moduli all greater than 1 (Problem 7), would make x^n + f(x) irreducible for infinitely many n for every integer f with f(0) != 0 and f(1) != -1.

Source: https://eudml.org/doc/204820.

Bears on. #7

Results to transcribe.

• Theorem 1: Proposition A (for every integer f with f(0) != 0, f(1) != -1, f not identically 1 there is an arithmetic progression N with x^v + f(x) irreducible for v in N) is equivalent to proposition B on finite covering systems, that some quotient m_j/m_i is a prime power q^a (a >= 0) with a_j != a_i mod m_i and either q > 2 or m_i odd or a_j != a_i mod (m_i/2).
• Theorem 2: C implies B and B implies D, where C says every finite covering system with all moduli > 1 has two equal moduli or an even modulus, and D says every such system has at least one modulus dividing another.
• Consequence of the Selfridge conjecture: If no finite covering system has distinct odd moduli greater than 1, then for every integer polynomial f with f(0) != 0 and f(1) != -1, x^n + f(x) is irreducible for infinitely many n.
• Lemma 1: If monic F_1,...,F_r in Z[x] are pairwise relatively prime modulo every prime, then there is f in Z[x] with f = a_i mod F_i for each i and deg f < deg(product of F_i).
• Lemma 2: For q prime and m <= n, the cyclotomic polynomials X_m and X_n are relatively prime mod q except when n/m = q^a (a >= 0), in which case X_n = X_m^{phi(n)/phi(m)} mod q.