Hilbert genus fields of some number fields with high degrees

TL;DR

This paper investigates properties of Hilbert genus fields and constructs these fields for certain number fields with high degrees, specifically those generated by roots of unity and square roots of specific integers.

Contribution

It provides new insights into the structure of Hilbert genus fields for complex number fields with high degrees, including explicit constructions for fields defined by roots of unity and specific square-free integers.

Findings

Properties of Hilbert genus fields are characterized.

Explicit construction methods for Hilbert genus fields of specified number fields.

Results extend understanding of genus fields in high-degree number fields.

Abstract

The aim of this paper is to give some properties of Hilbert genus fields and construct the Hilbert genus fields of the fields $L_{m,d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$, where $m\geq 3$ is a positive integer and $d$ is a square-free integer whose prime divisors are congruent to $\pm 3\pmod 8$ or $9\pmod{16}$.