Exercise 11.1.6

Proof. Write $\bar{F},\bar{\phi },\bar{\psi }\in {𝔽}_p[x]$ the reductions modulo $p$ of $F,\phi ,\psi \in \mathbb{Z}[x]$.

Let $F\in \mathbb{Z}[x]$ be a polynomial of degree $n>0$, and assume that the leading coefficient of $F$ is not divisible by $p$, so $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\bar{F})=n>0$, $\bar{F}$ is not zero, and $\bar{F}$ is not an unit of ${𝔽}_p[x]$.

$\text{∙}$

Suppose that there exist polynomials $\phi ,\psi ,\chi \in \mathbb{Z}[x]$, where $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\phi ),\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\psi )<n$, such that $$\phi (x)\psi (x)=F(x)+\mathit{p\chi }(x).$$

Then $\bar{\phi }(x)\bar{\psi }(x)=\bar{F}(x)$, and $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\bar{\phi })\le \mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\phi )<n$, and $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\bar{\psi })\le \mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\psi )<n$, with $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\bar{F})\ge 1$. Hence $\bar{F}$ is not irreducible in ${𝔽}_p[x]$.

Conclusion: If $\bar{F}$ is irreducible over ${𝔽}_p$, it is not possible to find polynomials $\phi ,\psi ,\chi \in \mathbb{Z}[x]$, where $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\phi ),\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\psi )<n$, such that

$$\phi (x)\psi (x)=F(x)+\mathit{p\chi }(x).$$

$\text{∙}$

Conversely, suppose that $\bar{F}$ is not irreducible in ${𝔽}_p[x]$. Then there exist polynomials $p,q\in {𝔽}_p[x]$ with $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(p),\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(q)<n$ and $\bar{F}(x)=p(x)q(x)$.

There exist some polynomials $\phi ,\psi \in \mathbb{Z}[x]$ such that $\bar{\phi }=p,\bar{\psi }=q$ and $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\phi )=\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(p)<n,\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\psi )=\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(q)<n$ (if $p={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=0}^d[a_i]x^i$, with $[a_d]\ne [0]$, take $\phi ={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=0}^d{a}_i{x}^i$). Thus $\bar{F}=\bar{\phi }\bar{\psi }$, therefore $\overline{\mathit{\phi \psi }-F}=\bar{0}$, so $\mathit{\phi \psi }-F\in \mathit{p\mathbb{Z}}[x]$: there exists $\chi \in \mathbb{Z}[x]$ such that

$$\phi (x)\psi (x)=F(x)+\mathit{p\chi }(x).$$

$\bar{F}$ is irreducible over ${𝔽}_p$ if and only if it is not possible to find polynomials $\phi ,\psi ,\chi \in \mathbb{Z}[x]$, where $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\phi ),\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(\psi )<n$, such that

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