Equation of the line in an affine plane over a polynomial field What are some examples of this? Say for $F_{4}$. I know this is a very simple question, but I can't find any info on it.
Edit: Yes, I was thinking of $F_{2}[x]/(x^2+x+1)$. I was confused.
 A: Ok. So $F_4=\{0,1,\alpha,\alpha+1\}$, where $\alpha$ is a root of the equation $\alpha^2=\alpha+1$. Or, using the notation of quotient rings, $\alpha=x+ I\in F_2[x]/I$, where $I$ is the ideal generated by $x^2+x+1$.
About lines over $F_4$? The recipe from anon's comment applies to any field. A line is the set of solutions $(x,y)$ of a linear equation
$$
ax+by=c,
$$
where $a,b,c$ are constants from that field, and at least one of $a,b$ is non-zero.
If here $b\neq0$, then we can solve $y$ from this equation, and express the line in the
'high-school form'
$$
y=-\frac{a}{b}x+\frac{c}{b}
$$
with the 'slope' $-a/b$ and 'intercept' $c/a$ visible. OTOH, if $b=0$, then we assumed that $a\neq0$, and (solving $x$) the equation is equivalent to
$$
x=\frac{c}{a}
$$
that represents a line parallel to the $y$-axis.
As an example of a line over $F_4$ let us look at the following equation
$$
\alpha x+\alpha^2y=1.
$$
Because $\alpha^2=\alpha+1\neq0$, we can solve for $y$. To that end we need the inverse
of $\alpha^2$. Here we get
$$\alpha^3=\alpha\cdot\alpha^2=\alpha(\alpha+1)=\alpha^2+\alpha=(\alpha+1)+\alpha=2\alpha+1=1$$
by repeatedly applying the equation $\alpha^2=\alpha+1$. So $\alpha^2\cdot\alpha=1$, and we can infer that $\alpha=(\alpha^2)^{-1}$ is the inverse. Let's multiply our equation by $\alpha$! We get
$$
\alpha\alpha x+\alpha\alpha^2 y=\alpha 1\Leftrightarrow \alpha^2x+y=\alpha.
$$
This gives us the equation of the line
$$
y=-\alpha^2x+\alpha=\alpha^2x+\alpha=(\alpha+1)x+\alpha
$$
in the usual 'slope and intercept' -form. 
We can also list all the four points on this line. We get the $y$-coordinate of a point from this equation by plugging in all four elements of $F_4$ as the $x$. The points are (leaving this as an exercise in $F_4$-arithmetic:
$$
(0,\alpha),\ (1,1),\ (\alpha,\alpha+1),\ (\alpha+1,0).
$$
