  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.
