When defining a vector space is the scalar part of the field or always a real number I've stumbled upon an exercise that takest the set of integers $\Bbb{Z}$, defines addition and multiplication as usual but scalar multiplication as $\lfloor{\alpha}\rfloor * k$, where $\alpha$ is the scalar and $k$ the element of the vector space and proceeds to claim that this set is not a vector space.
Wikipedia says the scalar is in field $\Bbb{F}$:
In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F. 

which in this case is the integers $\Bbb{Z}$?
However, the solution to the exercise only makes sense if the scalar is in $\Bbb{R}$. What am I misunderstanding?
 A: In your case, $\mathbb Z$ plays the role of the set $V$ defined in Wikipedia article.
And you're right, the scalar multiplication defined in your exercise makes sense if the field $\mathbb F$ is the field of the reals $\mathbb R$.
So here a vector is just an integer in $\mathbb Z$.
A: Let $E$ be an abelian group and $\mathbb{k}$ be a field. Let $\left(\lambda,v\right)\in \mathbb{k}\times E \mapsto \lambda\cdot v \in E$ be a function. We say that $(E,~\cdot~)$ is a vector space if $\forall \lambda, \mu, v,w$
\begin{align}
\lambda\cdot(\mu \cdot v) &=(\lambda\mu)\cdot v &
 1_{\mathbb{k}}\cdot v &= v \\
\lambda\cdot(v+w)&= \lambda\cdot v + \lambda\cdot w &(\lambda+\mu)\cdot v &=\lambda\cdot v + \mu\cdot v
\end{align}
These four assumptions just say that the group structure of $E$ and the field structure of $\mathbb{k}$ are compatible.
In your question, $E = \mathbb{Z}$ is an abelian group and $\mathbb{k}=\mathbb{R}$ is a field, and $\lambda \cdot v = \lfloor{\lambda}\rfloor v$. To show that $(\mathbb{Z}, ~\cdot~)$ is not a vector space over $\mathbb{R}$, you can show that one of the four assumptions above is not respected.
For example, let $v = 1$ and $\lambda = \mu = \frac12$. Then :
\begin{align}
\left(\lambda + \mu\right) \cdot v = (1)\cdot 1 = \lfloor 1\rfloor \times 1 &= 1 \\
\lambda\cdot v + \mu \cdot v = \lfloor \frac12 \rfloor\times 1 +\lfloor \frac12 \rfloor\times 1 &= 0
\end{align}
so for this particular choice of $\lambda,\mu,v$, we have $(\lambda+\mu)v \neq \lambda v + \mu v$. Thus, $\mathbb{Z}$ with this operation is not a vector field over $\mathbb{R}$.
