The number of one-dimensional vector spaces in a field

Question

Let $p$ be a prime number, $F=F_p$ a field with $p$ elements. $V$ is a vector space, $n$-dimensional over $F$. Calculate the number of one-dimensional vector spaces in $V$.

I tried to solve it, but without a success, I can't use Burnside's lemma because we didn't prove it. I would like to get help with this question.

Thanks

Answer 1

Hints:

== Every $\;1$-dimensional subspace of $\;V\;$ is of the form ${}$ Span$\,\{v\}\;,\;\;0\neq v\in V\;$

== There are exactly $\;p-1\;$ vectors in $\;U:=$Span $\,\{v\}\;$ for which their span equals $\;U\;$ itself.

Answer 2

The number of elements in $V$ is $p^n$. An one-dimensional subspace has $p$ elements. The intersection of two different one-dimensional subspaces has only one element ($0$), and every non-zero element generates an one-dimensional subspace.

Then, if $k$ is the number of such subspaces: $$p^n=kp-(k-1)=1+k(p-1)$$

Therefore, $$k=\frac{p^n-1}{p-1}=\sum_{j=0}^{n-1}p^j$$