Let $p$ be a prime number and $q$ a power of $p$. Let $K$ be an algebraic closure of $\mathbb{F}_{q}$. Say that a polynomial $f(X) \in K[X]$ is a $q$-polynomial if it is of the form
$$f(X) = \sum_{i=0}^{n} a_{i} X^{q^{i}}$$
Let $f(X)$ be a $q$-polynomial of degree $q^{n}$, with $a_{0} \neq 0$. Show that the set of zeros of $f(X)$ is an $n$-dimensional vector-space over $\mathbb{F}_{q}$.