What question did this study set out to answer?

To establish a complete Galois-theoretic and arithmetic classification of splitting fields for power-compositional polynomials constructed from real and imaginary quadratic units.

August 19, 2026Open Access

Galois Groups of Powered Quadratic Unit Polynomials: Real Kummer Structure and Imaginary Cyclotomic Classification

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DBDavid Betzer

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Implication

Mathematical analysis reveals the splitting-field and Galois structures of power-compositional quadratic unit polynomials, highlighting divergent Kummer and cyclotomic behaviors across real and...

Key Points

To establish a complete Galois-theoretic and arithmetic classification of splitting fields for power-compositional polynomials constructed from real and imaginary quadratic units.
Analyzed the polynomial family Q_{\varepsilon,\ell}(x) = (x^\ell - \varepsilon)(x^\ell - \bar{\varepsilon}) across quadratic fields F/\mathbb{Q} and integer exponents \ell \ge 2.
Characterized field extensions via Kummer theory, cyclotomic composita, resultant-derived discriminants, and Dirichlet's unit theorem.
Constructed explicit concrete models using the golden-ratio unit along with Gaussian and Eisenstein imaginary quadratic cases.
Splitting fields for general real quadratic units achieve degree 2\ell \cdot \varphi(\ell) with the Galois group structure (\mathbb{Z}/\ell\mathbb{Z} \rtimes (\mathbb{Z}/\ell\mathbb{Z})^\times) \rtimes \mathbb{Z}/2\mathbb{Z}.
The polynomial discriminant evaluates explicitly to disc(Q) = \ell^{2\ell} \cdot N(\varepsilon)^{\ell-1} \cdot \Delta^\ell, identifying full finite ramification support.
Imaginary quadratic units collapse completely to cyclotomic field extensions with abelian Galois groups of degree \varphi(n) due to unit rank zero.

Cite This Study

David Betzer (2026) studied this question.

synapsesocial.com/papers/6a85642503308d306e2d7c01 https://doi.org/10.5281/zenodo.21986780

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Implication

Mathematical analysis reveals the splitting-field and Galois structures of power-compositional quadratic unit polynomials, highlighting divergent Kummer and cyclotomic behaviors across real and...

Galois Groups of Powered Quadratic Unit Polynomials: Real Kummer Structure and Imaginary Cyclotomic Classification

Discussion

Member takes

Implication

Key Points

Cite This Study

Galois Groups of Powered Quadratic Unit Polynomials: Real Kummer Structure and Imaginary Cyclotomic Classification

Discussion

Member takes

Implication

Key Points

Cite This Study

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