Notes on the Quadratic Integers and Real Quadratic Number Fields

Abstract

It is shown that when a real quadratic integer $\xi$ of fixed norm $\mu$ is considered, the fundamental unit ${\epsilon }_d$ of the field $\mathbb{Q}(\xi )=\mathbb{Q}(\sqrt{d})$ satisfies $\log {\epsilon }_d\gg (\log d{)}^2$ 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.