The construction of the Hilbert genus fields of real cyclic quartic fields

TL;DR

This paper constructs the Hilbert genus fields for a class of real cyclic quartic fields generated by specific algebraic expressions involving fundamental units and square-free integers.

Contribution

It provides an explicit construction of the Hilbert genus fields for these real cyclic quartic fields, extending understanding of their algebraic and number-theoretic properties.

Findings

Explicit formulas for Hilbert genus fields of the specified quartic fields

Characterization of the structure of these genus fields

Extension of known results to new classes of number 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. In the present paper, we construct the Hilbert genus field of the real cyclic quartic fields $\mathbb{Q}(\sqrt{a\varepsilon_p\sqrt{p}})$.