论文标题
在多项式的普通素数上
On the Common Prime Divisors of Polynomials
论文作者
论文摘要
带有整数系数的多项式$ p $的主要除数是$ p（x）\ equiv 0 \ pmod {p} $的primes $ p $。我们的主要结果是，任何几个多项式的共同质量分节恰好是某些单个多项式的质数。通过将此结果与AX的定理相结合，我们可以获得具有整数系数的任何系统$ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ as可解决的模元$ p $的primes $ p $。此外，我们证明了多项式主要分裂的密度结果。该文章是对代数数理论和GALOIS理论的光介绍。
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.
