Is a vector space over a finite field always finite?

Solution 1:

Yes to your comment below Ahriman's, Moose.

These are not the only examples, though: if $\,\Bbb F=\Bbb F_p\,$ is the prime finite field of order a prime $\,p\,$, then $\,\Bbb F\times \Bbb F\times...\,$ is an infinite vector space over $\,\Bbb F\,$.

In short: a vector space over a finite field is finite iff it is finite dimensional.

Solution 2:

The direct sum $F=\bigoplus_{i\in I} \Bbb F$ is a vector space over $\Bbb F$ of dimension equal to the cardinality of I. Thus you can get a vector field of any dimension, for every field.

Solution 3:

$\frac{\mathbb{Z}}{2\mathbb{Z}}[X]$ is a vector space but is not finite since it contains $1, X, X^2, ...$

Solution 4:

If a commutative ring $R$ with unity contains a field $F$, then $R$ is a vector space over $F$. Thus it suffices to find an infinite commutative ring with unity that has a finite field as a subfield. One example is $F[x]$ for any finite field $F$.