What does it mean for a polynomial with integer coefficients to have a root in $p$-adic integers? I have just begun delving into $p$-adic number theory. I was wondering, given a poynomial $f(x)$ with integer coefficients, what does it mean when we say, $f(x)$ has a root in $\mathbb{Z}_2$, for instance.
If $f(x) = f_0 + f_1 x + \ldots + f_d x^d \in \mathbb{Z}[x]$, and $f(r) = 0$ over $\mathbb{Z}_2$ then $$f_0 + f_1 r + \ldots + f_d r^d \equiv 0 \bmod 2.$$ Equivalently, $f(x) \equiv (x-r) g \bmod 2$ for some polynomial $g(x) \in \mathbb{Z}_2[x].$
If $f(x) = f_0 + f_1 x + \ldots + f_d x^d \in \mathbb{Z}[x]$, and $f(r) = 0$ over $\mathbb{Z}_2$ then $$f_0 + f_1 r + \ldots + f_d r^d \equiv 0 \bmod 2.$$ Equivalently, $f(x) \equiv (x-r) g \bmod 2$ for some polynomial $g(x) \in \mathbb{Z}_2[x].$