tensor product of a vector space and finite field

Question

I want to know how we can interpret and define the tensor product of V as a vector space with a F as a finite field?

Answer 1

For any field $K$ and $K$ vector space $V$, we have that $V\cong V\otimes_K K$ via the mapping $v\mapsto v\otimes 1$.

This mapping is made possible because all of the elements of $V\otimes_K K$ can be written as $x\otimes 1$. You get that $v\otimes f=vf\otimes 1$ for any simple tensor, and then for a general element $\sum \alpha_i(v_i\otimes \beta_i)=\sum (\alpha_i\beta_i v_i\otimes 1 )=\sum (\alpha_i\beta_i v_i)\otimes 1$.

More generally, $V\otimes_K K^n\cong V^n$. In particular, if $F$ is a finite dimensional extension of $K$, you know that $F\cong K^t$ for some positive integer $t$, and then

$V\otimes_K F\cong V\otimes_K K^t\cong V^t$