Euclidean polynomials for certain arithmetic progressions and the multiplicative group of Fp2

Abstract

Let f(x) be a polynomial with integer coefficients. We say that the prime p is a prime divisor of f(x) if p divides f(m) some integer m. For each positive integer n, we give an explicit construction of a polynomial all of whose prime divisors are ±1 modulo (8n+4). Consequently, this specific polynomial serves as an “Euclidean" polynomial for the Euclidean proof of Dirichlet's theorem on primes in the arithmetic progression ±1 (mod 8n+4): Let F_p² be a finite field with p² elements. We use that the multiplicative group of F_p² is cyclic in our proof.