Definition (Polynomials)

Let $\mathbb{F}$ be a field and consider the ring $\mathbb{F}[x]$ of polynomials over $\mathbb{F}$. Explicitly this is the set $$\mathbb{F}[x]=\{ a_{0}+a_{1}x+\dots+a_{n}x^{n}:a_{i}\in\mathbb{F},n\ge0 \}$$

Definition (Monic)

We say a polynomial $f(x)=a_{n}x^{n}+\dots+a_{1}x+a_{0}$ of degree $n$ is monic if $$a_{n}=1$$

Definition (Nonzero constant polynomials)

The Nonzero Constant are a class of functions $f\in\mathbb{F}[x]$ s.t. $$\deg f=0\text{ and }f|g \ \ \forall g\in\mathbb{F}[x]$$ These are the units of $\mathbb{F}[x]$.

Definition (Associate)

Let $f,g\in\mathbb{F}[x]$ be polynomials. We say $f$ and $g$ are associates if $\frac{f}{g}$ is a unit, i.e. $$\deg \frac{f}{g}=0$$ $\frac{f}{g}$ is a nonzero constant polynomial.

Definition (Irreducible)

A polynomial $f\in\mathbb{F}[x]$ is said to be irreducible if $f$ is NOT divisible by any $g\in\mathbb{F}[x]$ where $$1\le\deg g<\deg f$$

Definition (Relatively prime)

Given two polynomials $a$ and $b$, they are said to be relatively prime if their gcd is a unit.