Completely normal elements in finite abelian extensions

TL;DR

This paper constructs explicit completely normal elements in specific number field extensions, including cyclotomic fields and modular function fields, providing new examples beyond previous known cases.

Contribution

It introduces new explicit completely normal elements in maximal real subfields of cyclotomic fields and certain modular function field extensions, expanding the known classes of such elements.

Findings

Constructed a completely normal element in the maximal real subfield of a cyclotomic field.

Identified a completely normal element in an extension of modular function fields.

Extended the criterion for normal elements to new classes of number fields.

Abstract

We give a completely normal element in the maximal real subfield of a cyclotomic field over the field of rational numbers, which is different from that of Okada. This result is a consequence of the criterion for a normal element developed in [Normal bases of ray class fields over imaginary quadratic fields, Math. Zeit.]. Furthermore, we find a completely normal element in certain extension of modular function fields in terms of a quotient of the modular discriminant function.