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October 2016 Notes on quadratic integers and real quadratic number fields
Jeongho Park
Osaka J. Math. 53(4): 983-1002 (October 2016).

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.

Citation

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Jeongho Park. "Notes on quadratic integers and real quadratic number fields." Osaka J. Math. 53 (4) 983 - 1002, October 2016.

Information

Published: October 2016
First available in Project Euclid: 4 October 2016

zbMATH: 06654659
MathSciNet: MR3554852

Subjects:
Primary: 11R29
Secondary: 11J68, 11R11, 11Y40

Rights: Copyright © 2016 Osaka University and Osaka City University, Departments of Mathematics

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Vol.53 • No. 4 • October 2016
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