Irreducible Criterions in Q[x]

In this section, we will assume the base field is $\mathbb{Q}$, and we will discuss the irreducibility criterion for polynomials in $\mathbb{Q}[x]$.

The units in $\mathbb{Q}[x]$ are the non-zero constants, so we can always rescale:

$$f(x)\to N\cdot f(x)\text{where }N\in \mathbb{Z}$$

to make all the coefficients of $f(x)$ integers, the process does not change the irreducibility of $f(x)$, so in this section we will only consider polynomials with integer coefficients, but view them as polynomials in $\mathbb{Q}[x]$.

Rational Roots Theorem

First, we would like to use the root test at least for degree 2 or 3 polynomials. The following theorem tells us that only finite choices of rational numbers need to be checked.

Theorem:

Given a polynomial $f(x)=a_n{x}^n+\cdots +a_{1}x+a_0\in \mathbb{Z}[x],a_n\ne 0$, if $f(x)$ has a rational root, then the root must be of the form $\frac{p}{q}$ where $p\mid a_0$ and $q\mid a_n$.

We can use the Rational Roots Theorem to show that $f(x)=x^3-3x+1$ is irreducible in $\mathbb{Q}[x]$:

1. The Rational Roots Theorem tells us that the only possible rational roots of $f(x)$ are $\pm 1$. We can check that $f(1)=f(-1)=1\ne 0$, so $f(x)$ has no rational roots.
2. By root test in degree 3, $f(x)$ is irreducible in $\mathbb{Q}[x]$.

Eisenstein’s Criterion

Eisenstein’s criterion is a powerful tool to determine the irreducibility of a polynomial. Unlike root test, Eisenstein’s criterion does not require us to find the roots of the polynomial, and is applicable to polynomials of any degree.

The proof of Eisenstein’s criterion will need some tricks in elementary number theory, so we will only state the criterion here.

Theorem:

Given a polynomial $f(x)=a_n{x}^n+\cdots +a_{1}x+a_0\in \mathbb{Z}[x],a_n\ne 0$, if there exists a prime $p$ such that:

1. $p\mid a_i$ for all $i\ne n$.
2. $p^2\nmid a_0$.
3. $p\nmid a_n$. then $f(x)$ is irreducible in $\mathbb{Q}[x]$.

We show following examples to illustrate the power of Eisenstein’s criterion.

1. $f(x)=x^4+2x^2+2x+2$ is irreducible in $\mathbb{Q}[x]$ by Eisenstein’s criterion with $p=2$.
2. $f(x)=x^n+p$ is irreducible in $\mathbb{Q}[x]$ by Eisenstein’s criterion with $p$ prime, and $n>1$.