Modular Arithmetic/Lagrange's Theorem

Lagrange's Theorem

If $p$ is a prime number, $x\in \mathbb {Z} /p\mathbb {Z}$, and $f(x)\in {\mathbb {Z} [x]}$ is a polynomial with integer coefficients, then either:

Every coefficient of $f(x)$ is divisible by $p$,

or

$f(x)\equiv 0{\pmod {p}}$ has at most $\deg f(x)$ solutions.

Where $\deg f(x)$ is the degree of $f(x)$. If the modulus is not prime, then it is possible for there to be more than $\deg f(x)$ solutions.

(WIP)