How many elements are in the projective line $\mathbb{P}^{1}(k)$ if k is a finite field

Question

Assume k is a finite field with n elements, how many elements are in the projective line $\mathbb{P}^{1}(k)$ and how do I work this out?

I know that an element of $\mathbb{P}^{1}(k)$ is represented by $[a, b]$, where $a, b \in k$, not both of the coordinates are 0, and two elements $[a, b]$ and $[c, d]$ are equal if for some $\lambda \in k^{*}$ we have $a=\lambda c, b=\lambda d$

However, I’m not sure how I can use this to work out the number of elements?

Likewise how would I advance this to work out the number of elements in $\mathbb{P}^{2}(k)$ where the elements are the triples [a,b,c] ?

Answer 1

The elements are (in homogeneous coordinates):

$$(0,1),(1,1),\ldots, (q-1,1), (1,0),$$

where $k = \{0,1,\ldots,q-1\}$ has $q$ elements. So the number of elements is $q+1$.

In the first $q$ elements, the 2nd coordinate is normalized to $1$. In the last element, the 2nd coordinate is $0$ and the first coordinate is normalized to $1$.

Using normalization, the projective plane can be described similarly.

Answer 2

Your description of a projective line can be paraphrased as follows:

The projective line over a field $k$ consists of all nonzero points in $k^2$, where two points are considered 'the same' if they are on the same line through the origin.

Then if $k$ is a finite field of $q$ elements, there are $q^2-1$ nonzero points in $k^2$. Each line through the origin contains $q-1$ nonzero points, so this gives $\frac{q^2-1}{q-1}$ distinct points for the projective line over $k$.

This also generalizes to projective space over finite fields; projective $d$-space over a finite field of $q$ elements contains precisely $\frac{q^{d+1}-1}{q-1}$ points.

Answer 3

If you want to think geometrically you can work it out this way.

The affine line over a field of cardinality $q$ is the one dimensional vector space over the field, so has $q$ elements. Add the point at infinity to construct the projective line with $q+1$ elements.

You can proceed inductively up the dimensions. The projective plane is the affine plane with a projective line at infinity, so it has $q^2 + q + 1$ points.

This is the lovely seven point projective plane on the two element field:

https://en.wikipedia.org/wiki/Projective_plane