Why is $\mathbb{Z}$ not a vector space? Why is the set of integers $\mathbb{Z}$ not a vector space over $\mathbb{Z}$? It seems to satisfy all of the necessary axioms. Scaling also works because we're only doing scaling of the integer (vector) by an integer (scalar), which produces another integer (vector).
I'm not quite satisfied with "vector spaces are be defined over fields and $\mathbb{Z}$ is not a field" because that simply punts the question to be "Why are we required to define vector spaces over fields?" Multiplicative inverses never come into play with vector spaces, so it seems like we only really need a subset of the field axioms (which $Z$ satisfies).
I'm looking for an answer at an advanced-undergraduate level (e.g. Axler's 'Linear Algebra Done Right'). I've seen this question, but I don't understand the proof.
 A: Not to restate the point, but as jjagmath mentioned, multiplicative inverses come into play often when considering vector spaces. You've said you're reading Axler; he writes about a linear dependence lemma. Specifically,
If $v_1,...,v_m$ is a linearly dependent list, then there is some $1\leq j \leq m$ such that $v_j \in $ span$\{v_1,...,v_{j-1}\}$. To go about proving this, you know that because the list is linearly dependent
$$
a_1v_1+...+a_m v_m = 0
$$
non trivially, and if you assume wlog that $a_j\neq 0$ you get that
$$
v_j = -\frac{a_1}{a_j}v_1-...-\frac{a_{j-1}}{a_j}v_{j-1}
$$
But this division implicitly requires that the field elements have multiplicative inverses! In fact, what makes a lot of the structure theory of vector spaces so "nice" (Vector spaces of same dimension are isomorphic, etc.) is the fact that we are over a field (with these inverses) rather than an arbitrary commutative ring.
Edit: You are however correct in saying that the vector space axioms themselves don't ever require multiplicative inverses. It turns out these axioms are the same for a module, which you can think of as a generalization of a vector space where fields are replaced by rings.
A: "Multiplicative inverses never come into play with vector spaces". That's completely false. Try to invert a matrix without multiplicative inverses.
The notion of module generalize vector spaces with rings instead of fields. And there do exists theorems that are valid in that general context, but there are a lot of very powerful theorems from Linear Algebra that don't generalize to modules.
