Splitting of integer polynomials over fields of prime order

Abstract

It is well known that a polynomial $\varphi (X)\in \mathbb{Z}[X]$ 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 $\varphi$(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 $\varphi$ 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 $\varphi$ 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 $\varphi$ is also proven.

Already well-known result