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Suppose that $V$ is vector space with dimension $p^2$ defined on a finite field $F$. How many subspaces of dimension one, $V$ has?

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You mean $V$ is a vector space, I think, and I suppose $F$ is a finite field. –  Robert Israel May 25 '12 at 19:59

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up vote 3 down vote accepted

Hint: each subspace of dimension $1$ is spanned by a nonzero vector. How many of those are there? How many vectors span the same subspace?

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I know that we have at most $q^2-1$ elements. But I was told that there are $q-1$ elements reiteratively. I don't know how this later occurs. –  Nancy Rutkowskie May 25 '12 at 20:06
    
How many elements does the field $F$ have? –  Robert Israel May 25 '12 at 20:25
    
I am so sorry. The Field has $p$ elements and I confused $p$ with $q$ in my previuos comment. –  Nancy Rutkowskie May 25 '12 at 21:00
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So there are $p^{p^2}$ vectors in $V$, of which all but $1$ are nonzero. Now, how many span the same $1$-dimensional subspace? –  Robert Israel May 25 '12 at 22:03

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