The image of the proof (in Axler's book) has been attached onto this post.
My question/issue with the proof is that it assumes that each of $a_i/a_j, i \in \{1, \dots, j - 1\}$ is another element in the set where $a_i$ comes from. In the book, it is assumed that coefficients $a_i \in F$, where $F$ is either $\mathbb{R}$ or $\mathbb{C}$; in this case, the proof works. But if $a_i \in \mathbb{Z}$, for example, then this proof runs into a very obvious issue.
I recall that earlier in the book, Axler notes that $F$ can represent any field, and since the set of integers $Z$ is not a field, this proof isn't meant to apply here?
$\mathbb{Z}$ is not a field, so it is not a counterexample. The field axioms guarantee that $a_i/a_j$ is back in $F$ for any $a_i,a_j\in F$ with $a_j\neq0$.