2020/06/01 by Olli Järviniemi, Järviniemi, Olli
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2006.00941
openalex publication_date 2020/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The prime divisors of a polynomial P with integer coefficients are those primes p for which P(x) ≡ 0 \pmodp is solvable. Our main result is that the common prime divisors of any several polynomials are exactly the prime divisors of some single polynomial. By combining this result with a theorem of Ax we get that for any system F of multivariate polynomial equations with integer coefficients, the set of primes p for which F is solvable modulo p is the set of prime divisors of some univariate polynomial. In addition, we prove results on the densities of the prime divisors of polynomials. The article serves as a light introduction to algebraic number theory and Galois theory.