Eisenstein's Criterion

Sometimes, we want to determine if a polynomial is irreducible without testing all possible factors. Thus, we have formulated some criteria to determine if a polynomial is irreducible.

Eisenstein's criterion

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial with integer coefficients and $p$ a prime number such that:

• $p\mid a_0,p\mid a_1,\cdots,p\mid a_{n-1}$,
• $p\nmid a_n$ and $p^2\nmid a_0$. Then $f(x)$ is irreducible in $\mathbb{Z}[x]$.

Now we prove it.

For the sake of contradiction, suppose that $f(x)$ is reducible. Then, let $f(x)=g(x)h(x)$ where $f(x)=\sum b_ix^i,\, g(X)=\sum c_jx^j$. Then $a_0=b_0c_0$. Since $p\mid a_0$ and $p^2\nmid a_0$ we may assume without loss of generality that $p\mid b_0$ and $p\nmid c_0$. From $a_1=b_0c_1+b_1c_0$ we obtain $p|b_1$. Continuing with the coefficients of $g(x)$ we obtain that $p$ divides the leading coefficient of $g(x)$. This is a contradiction, because then $p\mid a_n=b^ic^j$.