Osaka Journal of Mathematics

Notes on quadratic integers and real quadratic number fields

Jeongho Park

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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.

Article information

Osaka J. Math., Volume 53, Number 4 (2016), 983-1002.

First available in Project Euclid: 4 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R11: Quadratic extensions 11J68: Approximation to algebraic numbers 11Y40: Algebraic number theory computations


Park, Jeongho. Notes on quadratic integers and real quadratic number fields. Osaka J. Math. 53 (2016), no. 4, 983--1002.

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