A note on quadratic cyclotomic extensions

2022/10/07 by Marques, Sophie, Mrema, Elizabeth
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2210.03563

Abstract

This paper provides two characterizations of the primitive roots of unity in quadratic cyclotomic extensions over arbitrary fields. Firstly, we introduce a mapping from ℕ to ℕ crucial for describing these roots, closely tied to their order over the field. Secondly, for any prime p, we determine the maximal natural number n such that ζ_pⁿ defines a quadratic cyclotomic extension over the field F. This characterization is uniform across different fields, regardless of their characteristic, and applies to both odd and even primes.

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