A note on quadratic cyclotomic extensions

TL;DR

This paper characterizes primitive roots of unity in quadratic cyclotomic extensions over arbitrary fields, providing a uniform approach that applies to all primes and field characteristics.

Contribution

It introduces a new mapping to describe roots of unity and determines the maximal extension degree for quadratic cyclotomic fields over any base field.

Findings

A mapping from natural numbers to natural numbers crucial for root description

Maximal natural number for quadratic cyclotomic extension over any field

Uniform characterization applicable to all primes and field characteristics

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 $\mathbb{N}$ 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.