On the Common Prime Divisors of Polynomials

TL;DR

This paper investigates the prime divisors of polynomials, showing that common prime divisors of multiple polynomials correspond to prime divisors of a single polynomial, and explores their densities, linking algebraic number theory and Galois theory.

Contribution

It proves that common prime divisors of several polynomials are exactly the prime divisors of a single polynomial, and extends this to systems of polynomial equations.

Findings

Common prime divisors of multiple polynomials correspond to prime divisors of one polynomial.

For polynomial systems, primes where the system is solvable mod p are prime divisors of a univariate polynomial.

Results on the density of prime divisors of polynomials.

Abstract

The prime divisors of a polynomial $P$ with integer coefficients are those primes $p$ for which $P(x) \equiv 0 \pmod{p}$ 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.