Do all monic polynomials factor into monic irreducibles? Do all monic polynomials in $\mathbb{Z}/n\mathbb{Z}[x]$ where $n$ is prime factor into monic irreducibles? 
Just by thinking about these monic polynomials and how they only act on a finite number of elements makes me think that the assertion is true, because if $d$ is a root of a monic polynomial $f(x)$ then $f(x)= (x-d)g(x)$ where $g(x)$ is monic, and you could apply the same rule iteratively until you exhaust all roots. But I am not sure how to really prove this, it sounds really similar to one part of the fundamental theorem of arithmetic, but I can't find how to quantify a polynomial the same way you would an integer for an inductive proof. How would I go about starting a proof of this?
 A: If $F$ is a field, then $F[X]$ is a unique factorization domain, so every nonzero polynomial in $F[X]$ can be factored into a unique product of irreducibles (up to scaling by units). If $f \in F[X]$ is monic, then $f$ can be factored as $f_1 \cdot f_2 \cdots f_n$ for irreducible polynomials $f_1, \ldots, f_n \in F[X]$ whose leading coefficients $c_{1}, \ldots, c_{n} \in F$ must be units. Since $\prod_{i=1}^{n} c_{i} = 1$, we can multiply each $f_{i}$ by $c_{i}^{-1}$ to get $\tilde{f_{i}}$ monic irreducible such that $f = \tilde{f_{1}} \cdot \tilde{f_{2}} \cdots \tilde{f_{n}}$, since this amounts to multiplying $f$ by $\prod_{i=1}^{n} c_{i}^{-1} = (\prod_{i=1}^{n} c_{i})^{-1} = 1^{-1} = 1$. 
This can be specialized to your case $F = \mathbb{Z}/p\mathbb{Z}$ for $p$ prime. 
A: If $F$ is a field then any factorization of a monic polynomial in $F[x]$ can be made monic. Inded, if $f$ factors $p$ and has leading coefficient $a\neq 1$, then $a^{-1}\cdot f$ is monic and has the same roots as $f$, and hence also factos $p$.
Now, if $p$ is monic, $\prod f_i$ is any factorization of $p$ and $a_i$ is the leading coefficient of each $f_i$, then $\prod a_i=1$ so of course $\prod {(a_i)}^{-1}={\left(\prod a_i\right)}^{-1}=1$. It follows that $\prod \left(a_i^{-1}\cdot f_i\right)$ is a monic factorization of $p$.
More generally, if $p$ is a polynomial in $F[x]$ with leading coefficient $c$ then one can always factor $p$ as $c$ times a product of monic polynomials.
