A note on quadratic cyclotomic extensions

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.