A note on quadratic cyclotomic extensions

Sophie Marques and Elizabeth Mrema
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 3, Pages 576–589
DOI: 10.7546/nntdm.2026.32.3.576-589
Full paper (PDF, 288 Kb)

Details

Authors and affiliations

Sophie Marques
Department of Mathematical Sciences, Stellenbosch University
Stellenbosch, 7600, South Africa

Elizabeth Mrema
Department of Mathematics, University of Dar es Salaam
P.O. Box 35061, Tanzania

Abstract

Schinzel’s classification of radical extensions relies on the structure of quadratic cyclotomic extensions, which also play an important role in the study of cyclotomic extensions over general fields. In this paper, we identify properties and invariants associated with quadratic cyclotomic extensions that are relevant to these classification problems. We determine the possible minimal polynomials of primitive roots of unity generating quadratic extensions, describe the prime-power levels at which such extensions occur, and assemble the prime-power information into a global classification of quadratic cyclotomic fields. The treatment applies in arbitrary characteristic, with the usual restriction that, in positive characteristic, the orders of the roots of unity considered are prime to the characteristic.

Keywords

• Cyclotomic extensions
• Field extensions
• Roots of unity
• Quadratic extensions
• Radical extensions

2020 Mathematics Subject Classification

• 12F05
• 12E05
• 12E20
• 12E10
• 12F10

References

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Manuscript history

• Received: 11 March 2026
• Revised: 7 October 2026
• Accepted: 23 October 2026
• Online First: 26 October 2026

Copyright information

Ⓒ 2026 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).