  This is the first in a series of two papers that study monogenicity of number
rings from a moduli-theoretic perspective. Given an extension of algebras
$B/A$, when is $B$ generated by a single element $\theta \in B$ over $A$? In
this paper, we show there is a scheme $\mathcal{M}_{B/A}$ parameterizing the
choice of a generator $\theta \in B$, a "moduli space" of generators. This
scheme relates naturally to Hilbert schemes and configuration spaces. We give
explicit equations and ample examples.
