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 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.

Monogenity and Power Integral Bases: Recent Developments

“…• M. E. Charkani and A. Deajim [26] (see also A. Deajim and L. El Fadil [28]) x p − m over number fields • M. Sahmoudi and M. E. Charkani [148] considered relative pure cyclic extensions • A. Soullami, M. Sahmoudi and O. Boughaleb [150] x 3 n + ax 3 s − b over number fields • O. Boughaleb, A. Soullami and M. Sahmoudi [23] x p n + ax p s − b over number fields • H. Smith [152] relative radical extensions • S. K. Khanduja and B. Jhorar [138] give equivalent versions of Dedekind criterion in general rings • S. Arpin, S. Bozlee, L. Herr and H. Smith [5], [6] study monogenity of number rings from a modul-theoretic perspective • R. Sekigawa [149] constructs an infinite number of cyclic relative extensions of prime degree that are relative monogenic…”

“…• Z. Fran ȗsić and B. Jadrijević [63] calculated generators of relative power integral bases in a family of quartic extensions of imaginary quadratic fields • I. Gaál [65] showed that index form equations in composites of a totally real cubic field and a complex quadratic field can be reduced to absolute Thue equations • I. Gaál [68] showed that the index form equations in composites of a totally real field and a complex quadratic field can be reduced to the absolute index form equations of the totally real field • I. Gaál [66] considered generators of power integral bases in fields generated by monogenic trinomials of type x 6 + 3x 3 + 3a • I. Gaál [67] considered generators of power integral bases in fields generated by monogenic binomial compositions of type (x 3 − b) 2 + 1 • I. Gaál [70] gave an efficient method to determine all generators of power integral bases of pure sextic fields • I. Gaál and L. Remete [78] considered monogenity in octic fields of type K = Q( 4√ a + bi) • I. Gaál [69] determined "small" solutions of the index form equation in K = Q( 6 √ m), for −5000 < m < 0 such that x 6 − m is monogenic (1521 fields) Experience: 6 √ m is the only generator of power integral bases • I. Gaál [71] determined "small" solutions of index form equations in K = Q( 8 √ m), −5000 < m < 0 such that x 8 − m is monogenic (2024 fields) Experience: 8 √ m is the only generator of power integral bases, except for m = −1 • I. Gaál [72] extended [54] on monogenity properties of trinomials of type x 4 + ax 2 + b • I. Gaál [73] calculated generators of power integral bases in families of number fields generated by a root of monogenic quartic polynomials considered in [88] In [72], [73] the method described in Section 4.4 was used, in [63], [78], [71], its relative analogue, see [76], [64].…”

Monogenity and Power Integral Bases: Recent Developments

“…Mostly, Dedekind's criterion is used. [106,107] studied monogenity of number rings from a modul-theoretic perspective; • R. Sekigawa [108] constructed an infinite number of cyclic relative extensions of prime degree that are relative monogenic.…”