A class number problem for imaginary cyclic number fields of 2-power degrees

TL;DR

This paper generalizes a 2024 result by proving that the class number of certain imaginary cyclic number fields of 2-power degrees cannot be a prime congruent to 3 mod 4, extending the understanding of class number properties.

Contribution

It extends previous work by proving non-primality of class numbers for a broader class of imaginary cyclic number fields of 2-power degrees.

Findings

Class numbers of these fields are never prime p ≡ 3 mod 4.

The result applies to non-quadratic imaginary cyclic fields.

Generalizes previous specific cases to a wider class.

Abstract

In 2024, M. K. Ram proved that the class number of an imaginary cyclic quartic number field is never equal to a prime $p\equiv 3\pmod 4$. Here we greatly generalize this result to the case of the non-quadratic imaginary cyclic number fields of $2$-power degrees and not necessarily prime class numbers.