Notes on 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 $\varepsilon_{d}$ 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.