Relation of F and V in a Vector Space In many books and on Wikipedia a vector space is defined as a tuple $(F, +, V)$ where $F$ is a field and $V$ an abelian Group plus some axiums that must hold which I will omit here. 
I also often see the phrase 'a vector space over $F$'. 
What I am missing in all definitions that I have come across it the relationship between $F$ and $V$. 
So my question is this:
How are the elements of $V$ related to $F$? Is it a general requirement that $V=F^n$? That is does each component $x_i$ of a vector $x \in V$ with $i \in n$ have to satisfy $x_i \in F$? Obviously thats the case for the common real and complex vector spaces $\mathbb R^n$ and $\mathbb C^n$, but is it also that way in the general case?
If not, can you give an example of a vector space where this not the case?
 A: The definition of the field $F$ is essential in defining the properties of the vector space. The same set $V$ of vectors, is a different vector space over different  fields $F$.
As a dramatic example, consider $\mathbb{R}$ as the set of vectors, with the usual addition, than  we can think at $\mathbb{R}$ as a vector space over $\mathbb{R}$ and in this case it has dimension $1$. But, it can be also a vecor space over the field $\mathbb{Q}$ of rational numbers, and in  this case it is a vector space of uncountable infinite dimension. (See Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?).
Another, less sophisticated example is $\mathbb{C}$. It is a vector space of dimension $1$ over $\mathbb{C}$ and a vector space of dimension $2$ over $\mathbb{R}$ (and it has uncountable infinite dimension as a vector space over $\mathbb{Q}$).
About the last question: note that the components of a vector are defined with respect to some basis and are the coefficients of a linear combination, so they are elements of the field.
