EUCLIDEAN POLYNOMIALS FOR CERTAIN ARITHMETIC PROGRESSIONS AND THE MULTIPLICATIVE GROUP OF F-p2

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 p(2) elements. We use that the multiplicative group of F-p2 is cyclic in our proof.