Splitting of integer polynomials over fields of prime order

TL;DR

This paper proves that for any polynomial over integers, there are infinitely many primes where it fully splits into linear factors over finite fields, using elementary number theory and field theory.

Contribution

It establishes the existence of infinitely many primes where a polynomial splits completely over finite fields, and characterizes this splitting in terms of roots in those fields.

Findings

Existence of infinitely many primes where the polynomial splits completely.

Splitting in finite fields is equivalent to having a root in the field for large primes.

Characterization of polynomials that split in finite fields based on roots in those fields.

Abstract

It is well known that a polynomial $\phi(X)\in \mathbb{Z}[X]$ of given degree $d$ factors into at most $d$ factors in $\mathbb{F}_p$ for any prime $p$. We prove in this paper the existence of infinitely many primes $q$ so that the given polynomial $\phi$(X) splits into exactly $d$ linear factors in $\mathbb{F}_q$ by using only elementary results in field theory and some elementary number theory by proving that $\phi$ splits in $\mathbb{F}_q$ iff $P$ has a root in $\mathbb{F}_q$ for all sufficiently large primes $q$, where $P\in \mathbb{Z}[X]$ is any polynomial such that $P$ has a root $\beta \in \mathbb{C}$ for which $\mathbb{Q}(\beta)$ is the splitting field of $\phi$ over $\mathbb{Q}$. Furthermore, we prove that any such $P$ splits in $\mathbb{F}_r$ iff it has a root in $\mathbb{F}_r$, for all sufficiently large primes $r$. Existence of infinitely many such $P$ for any given $\phi$ is also proven.