Notes on the Quadratic Integers and Real Quadratic Number Fields

TL;DR

This paper investigates properties of real quadratic number fields generated by quadratic integers of fixed norm, showing that their fundamental units grow rapidly with the discriminant and providing explicit constructions for related sets of radicands.

Contribution

It introduces a new construction for sets of radicands of real quadratic fields with fixed norm quadratic integers, and analyzes the growth of fundamental units in these fields.

Findings

Fundamental units satisfy  \, ext{log} \, \, ext{varepsilon}_d \, \, ext{ extgreater} \, ( ext{log} \, d)^2 almost always.

Construction of radicands via quadratic sequences captures all fields with certain properties.

Provides explicit estimates for the efficiency of the radicand construction.

Abstract

It is shown that when a real quadratic integer $\xi$ of fixed norm $\mu$ is considered, the fundamental unit $\varepsilon_d$ of the field $\mathbb{Q}(\xi) = \mathbb{Q}(\sqrt{d})$ satisfies $\log \varepsilon_d \gg (\log d)^2$ almost always. An easy construction of a more general set containing all the radicands $d$ of such fields is given via quadratic sequences, and the efficiency of this substitution is estimated explicitly. When $\mu = -1$, the construction gives all $d$'s for which the negative Pell's equation $X^2 - d Y^2 = -1$ (or more generally $X^2 - D Y^2 = -4$) is soluble. When $\mu$ is a prime, it gives all of the real quadratic fields in which the prime ideals lying over $\mu$ are principal.