The scheme of monogenic generators I: representability

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 θ∈ B over A ? In this paper, we show there is a scheme ℳ_B/A parameterizing the choice of a generator θ∈ B , a “moduli space” of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.