# Proving the Eisenstein’s criteria

Polynomial $A = x^n + a_{n-1}x^{n-1} + .. + a_1 x + a_0$, where each $a_i$ is divisible by some number $p$, is irreducible in $\mathbb Q$ when $p^2$ does not divide $a_0$.
If A is not irreducible then there are some $F$ and $G$ such as $A = FG$.
But that means that $a_0 = f_0 g_0$, which means that $(p | f_0)$ XOR $(p | g_0)$ (since $p^2$ does not divide $a_0$).
Let’s assume that $p | f_0$. Now let’s take a map $Z[x] \rightarrow (Z/pZ)[x]$ (I believe this method is called homomorphism-reduction).
$[A] = x^n$, but $[G] = \cdots + [g_0]$, where $[g_0] \neq 0$.
That is possible only when $[G]$ is constant and $[F] = [A]/[G]$. Which means that $A$ is irreducible.

Note: here $[a]$ denotes $f(a)$ where $f$ is a homomorphism $Z[x] \rightarrow (Z/pZ)[x]$.