Editing Examples of vector spaces (section)

==Finite vector spaces==
Apart from the trivial case of a zero-dimensional space over any field, a vector space over a field ''F'' has a finite number of elements if and only if ''F'' is a [[finite field]] and the vector space has a finite dimension. Thus we have ''F''<sub>''q''</sub>, the unique finite field (up to [[isomorphism]]) with ''q'' elements. Here ''q'' must be a power of a [[prime number|prime]] (''q'' = ''p''<sup>''m''</sup> with ''p'' prime). Then any ''n''-dimensional vector space ''V'' over ''F''<sub>''q''</sub> will have ''q''<sup>''n''</sup> elements. Note that the number of elements in ''V'' is also the power of a prime (because a power of a prime power is again a prime power). The primary example of such a space is the coordinate space (''F''<sub>''q''</sub>)<sup>''n''</sup>.

These vector spaces are of critical importance in the [[representation theory]] of [[finite group]]s, [[number theory]], and [[cryptography]].