So if I have a polynomial $p(x,y)$ and define a curve $C$ based on $p$, what is a divisor? In the context I'm looking at (where I'm trying to learn about Goppa codes), in Joyner et al.'s "Applied Abstract Algebra", it's given as:
A divisor is simply a (formal) finite sum of points of $C(\bar{\mathbb{F}})$, where $\bar{\mathbb{F}}$ is an algebraic closure of $\mathbb{F}$.
I looked online and every resource I found assumed a lot more background knowledge than I have.
Are divisors just points that satisfies $p(x,y)=0$, but taken not just from $\mathbb{F}$, but the closure of $\mathbb{F}$? I.e., if $p(x,y)=x^2+y$, then $(i,1)$ would be a divisor, as would $(-i,1)$ and hence their sum $(0,2)$ is also a divisor?
Let $C$ be an algebraic curve. A collection of points $P_1,…,P_n$ in $C$ with assigned integer multiplicities $k_1,…,k_n$ is called a divisor on $C$ and it is denoted
$$ D=k_1P_1+...+k_nP_n. $$
And that is it, it is a formal sum, so as it is defined it does not have any immediate meaning. For instance, on your example we can define the divisors $D_1=(i,1), D_2=(−i,1), D_3=−(i,1)+2(−i,1)$ and so on.
You can simply assign integers to points of your curve and sum them in a formal way to define divisors on your curve.