On Hilbert genus fields of imaginary cyclic quartic fields

TL;DR

This paper explicitly determines the Hilbert genus fields of certain imaginary cyclic quartic fields constructed from quadratic fields with prime discriminants, advancing understanding of their algebraic structure.

Contribution

It provides an explicit description of the Hilbert genus fields for a class of imaginary cyclic quartic fields involving fundamental units and square-free integers.

Findings

Explicit formulas for Hilbert genus fields of the specified quartic fields

Characterization of these fields in terms of fundamental units and square-free integers

Enhanced understanding of the algebraic structure of these quartic fields

Abstract

Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let $\varepsilon_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. The main aim of this paper is to determine explicitly the Hilbert genus field of the imaginary cyclic quartic fields of the form $\mathbb{Q}(\sqrt{-a\varepsilon_p\sqrt{p}})$.