A representation of a group $G$ on a vector space $V$ over a field $K$ is a group homomorphism from $G$ to $GL(V)$, the general linear group on $V$. That is, a representation is a map $ \rho \colon G \to GL(V) \,\! $
such that $\rho(g_1 g_2) = \rho(g_1) \rho(g_2) , \qquad \text{for all }g_1,g_2 \in G $
Here $V$ is called the representation space and the dimension of $V$ is called the dimension of the representation.
In the case where V is of finite dimension $n$ it is common to choose a basis for V and identify $GL(V)$ with $GL (n, K)$ the group of $n$-by-$n$ invertible matrices on the field $K$.
An integral representation of $G$ is a map $\rho : G \to GL_n(\mathbb Z)$, what is the vector space $V$ (the representation space) in this case of integral representation? and what is the field $F$ that makes $V$ an $F$-vector space?