A note on quadratic cyclotomic extensions

This paper provides two characterizations of the primitive roots of unity in quadratic cyclotomic extensions over arbitrary fields. Firstly, we introduce a mapping from $\mathbb{N}$ to $\mathbb{N}$ 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 $\zeta_{p^n}$ 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.