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 $p^n$?
Number of vectors in an n-dimensional vector space.
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1Yes. (extra words to meet minimum) – 2012-02-06
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0Thank you very much buddy. – 2012-02-06
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0@BrandonCarter A handy trick I picked from Didier Piau is to insert several `$ $` into the comment to meet the minimum. – 2012-02-06
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1Hint: Use the canonical isomorphism (which is, in particular, a bijection of sets) from your vector space to $(\mathbb{Z}/(p))^n$. – 2012-02-06
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0@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. – 2012-04-01
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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 $x_i\in\{0,1,\dots,p-1\}$.