# Number of vectors in an n-dimensional vector space.

How many vectors are there in an $n$-dimensional vector space over the field $\mathbb{Z}/(p)$ (where $p$ is prime)? Would the answer be $p^n$?

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Yes. (extra words to meet minimum) – Brandon Carter Feb 6 '12 at 18:32
Thank you very much buddy. – Hardy Feb 6 '12 at 18:35
@BrandonCarter A handy trick I picked from Didier Piau is to insert several  into the comment to meet the minimum. – Sasha Feb 6 '12 at 18:35
Hint: Use the canonical isomorphism (which is, in particular, a bijection of sets) from your vector space to $(\mathbb{Z}/(p))^n$. – M Turgeon Feb 6 '12 at 18:36
@M Turgeon: there isn't a canonical such isomorphism, as an abstract vector space isn't equipped with a canonical ordered basis (an ordered basis being the same thing as an isomorphism with $(\mathbb{Z}/(p))^n$). But of course there does exist a basis for the vector space, hence such an isomorphism. – Brad Apr 1 '12 at 20:57

Yes,

$$\#(\mathbb{F}_p^{\,n})=(\#\mathbb{F}_p)^n=p^n.$$

The elements can be explicitly constructed as $n$-tuples $(x_1,\dots,x_n)$ with each $x_i\in\{0,1,\dots,p-1\}$.

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