  With this paper, we gain a better understanding of the set of near-field
structures on a fixed scalar group. If we were able to describe all near-field
structures on a fixed scalar group, we could describe all near-vector spaces.
The near-field structures induced by isomorphisms of canonical near-vector
spaces differ by quasi-multiplicative bijections while those induced by
isomorphisms of near-fields differ by multiplicative bijections. This reveals
one of the fundamental differences between linear algebra and near-linear
algebra. We find an explicit description of all the elementary near-vector
spaces. Significantly, we construct an addition $\boxplus$ on $\mathbb{Q}$ such
that $(\mathbb{Q},\boxplus, \cdot)$ is isomorphic to
$(\mathbb{Q}(\sqrt{-19}),+, \cdot)$. We also describe explicitly sufficient
conditions for such an isomorphism to exist for more general extensions of
$\mathbb{Q}$. Moreover, under extra conditions, we still describe those
structures on $(\mathbb{R}, \cdot)$, and $(\mathbb{C}, \cdot)$.
