How many lines are there in $A^2\left(\mathbb{F}_p\right)$ over the finite field $\mathbb{F}_q$ of $q$ elements?

Question

How many lines are there in $A^2\left(\mathbb{F}_p\right)$ over the finite field $\mathbb{F}_q$ of $q$ elements?

I'm a bit stuck with the question, my approach is the following:

Number of points * Number of lines through each point = $p^2⋅(1+p+p^2)$

Is it the right direction or I'm wrong?

Thank you!

Answer 1

A line is given in $\mathbb A^2$ by an equation $$ax+by+c=0; a,b,c\in \mathbb F_q$$ whre $a \text{ and }b$ are not both 0. Thus there are $q^3-q$ possibilities for $(a,b,c)$. Tho triples $(a,b,c), (\alpha,\beta,\gamma)$ represent the same line iff there exists some $\zeta \in \mathbb F_q, \zeta \ne 0$ such that $$a=\zeta \alpha, b=\zeta \beta, c=\zeta \gamma$$. There are $q-1$ possibilities for $\zeta.$ Thus a line corresponds to an equivalence class of allowable triples and each equivalence class contains $q-1$ members so the number of lines = number of equivalence classes =$$ \frac{q^3-q}{q-1} =q^2+q.$$

Answer 2

First, $1+p+p^2$ is not the number of lines through a given point in $\mathbb{F}_p^2$.

Second, (number of points) $\times$ (number of lines through each point) counts each line multiple times - in fact, it counts a given line once for every point on the line! Thus, to compensate, you must divide by the number of points on each line.

If you fix these two issues, your calculation will be correct.

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Here's an advanced POV. Every $d$-dimensional affine subspace of $\mathbb{F}_q^n$ is a coset of a $d$-dimensional vector subspace. Every $d$-dimensional vector subspace has $|\mathbb{F}_q^n/\mathbb{F}_q^d|=q^{n-d}$ cosets. The number of $d$-dimensional vector subspaces of $\mathbb{F}_q^n$ is given by the $q$-binomial coefficient $[\begin{smallmatrix} n \\ d\end{smallmatrix}]_q$ (there are numerous well-known derivations of this). In particular, the one-dimensional subspaces of $\mathbb{F}_q^n$ form what's called the projective space $\mathbb{F}_q\mathbb{P}^{n-1}$, and in the case of $n=2$ we call $\mathbb{F}_q\mathbb{P}^1$ the projective line. Every 1D subspace of $\mathbb{F}_q^2$ is spanned by a unique vector of the form $[\begin{smallmatrix}x\\1\end{smallmatrix}]$ ($x\in\mathbb{F}_q$) or else $[\begin{smallmatrix}1\\0\end{smallmatrix}]$, for a total of $q+1$ elements in $\mathbb{F}_q\mathbb{P}^1$.