ring theory | (parentheses)

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.