The Scheme of Monogenic Generators I: Representability
Abstract
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 MB/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.
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