Theorems on irreducible polynomials

From LeicesterIPSC
Jump to: navigation, search

This page gives results that can be used in deciding whether or not a polynomial is irreducible in ${\mathbb Q}[x]$.


  • Proposition

Let $F$ be a field and $f(x)$ be a polynomial in $F[x]$. Let $0 \neq a \in F$. Then $f(x)$ is irreducible in $F[x]$ if and only if $af(x)$ is irreducible in $F[x]$.

$\qquad\qquad$ Remark: This result is particularly useful when we have a polynomial $f(x)$ in ${\mathbb Q}[x]$ and we multiply by some suitable non-zero integer $a$ in order to consider the polynomial $af(x)$ with integer coefficients instead.


  • Theorem (Root Test in ${\mathbb Q}$)

Let $f(x) = a_0 + a_1x + \cdots + a_nx^n$ be a polynomial in ${\mathbb Z}[x]$ with $a_n \neq 0$. If $k/l \in {\mathbb Q}$ is a root of $f(x)$, with $k, l$ integers and $k/l$ written in its lowest form, then $k | a_0$ and $l | a_n$.


  • Theorem (Eisenstein's Criterion)

Let $f(x) = a_0 + a_1x + \cdots + a_nx^n$ be a polynomial in ${\mathbb Z}[x]$ with $n \geq 1$. If there is a prime integer $p$ such that $p | a_0, p | a_1, \dots , p | a_{n-1}$, $p \not | a_n$ and $p^2 \not | a_0$ then $f(x)$ is irreducible in ${\mathbb Q}[x]$.


  • Theorem (Modular Irreducibility Test)

Let $f(x) = a_0 + a_1x + \cdots + a_nx^n$ be a polynomial in ${\mathbb Z}[x]$ with $n \geq 1$. If there is a prime integer $p$ such that $p \not | a_n$ and for which the polynomial $\overline{a_0} + \overline{a_1}x + \cdots + \overline{a_n}x^n$ is irreducible in ${\mathbb Z}_p[x]$, then $f(x)$ is irreducible in ${\mathbb Q}[x]$.

$\qquad\qquad$ Remark: for an integer $a$, we denote by $\overline{a}$ the residue (or remainder) of $a$ modulo $p$. Note also that ${\mathbb Z}_p$ is a field since $p$ is prime.




Next section on Cyclotomic polynomials

Back to Chapter 2: Polynomials