Osaka Journal of Mathematics

Notes on quadratic integers and real quadratic number fields

Jeongho Park

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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.

Article information

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

Dates
First available in Project Euclid: 4 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1475601827

Mathematical Reviews number (MathSciNet)
MR3554852

Zentralblatt MATH identifier
06654659

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

Citation

Park, Jeongho. Notes on quadratic integers and real quadratic number fields. Osaka J. Math. 53 (2016), no. 4, 983--1002. https://projecteuclid.org/euclid.ojm/1475601827


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References

  • H. Cohen: A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer, Berlin, 1993.
  • H. Cohen and H.W. Lenstra, Jr.: Heuristics on class groups of number fields; in Number Theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Math. 1068, Springer, Berlin, 33–62, 1984.
  • J.H.E. Cohn: The length of the period of the simple continued fraction of $d^{1/2}$, Pacific J. Math. 71 (1977), 21–32.
  • H. Davenport and H. Heilbronn: On the density of discriminants of cubic fields, II, Proc. Roy. Soc. London Ser. A 322 (1971), 405–420.
  • P.G.L. Dirichlet: Vorlesungen über Zahlentheorie, Braunschweig: F. Vieweg und Sohn, 1894.
  • É. Fouvry and F. Jouve: Size of regulators and consecutive square-free numbers, Math. Z. 273 (2013), 869–882.
  • É. Fouvry and F. Jouve: A positive density of fundamental discriminants with large regulator, Pacific J. Math. 262 (2013), 81–107.
  • É. Fouvry and J. Klüners: On the negative Pell equation, Ann. of Math. (2) 172 (2010), 2035–2104.
  • A. Granville: $ABC$ allows us to count squarefrees, Internat. Math. Res. Notices (1998), 991–1009.
  • G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers, fifth edition, Oxford Univ. Press, New York, 1979.
  • C. Hooley: On the Pellian equation and the class number of indefinite binary quadratic forms, J. Reine Angew. Math. 353 (1984), 98–131.
  • E.L. Ince: Cycles of Reduced Ideals in Quadratic Fields, Mathematical Tables 4, Cambridge Univ. Press, Cambridge, 1968.
  • M.J. Jacobson, Jr.: Experimental results on class groups of real quadratic fields (extended abstract); in Algorithmic Number Theory (Portland, OR, 1998), Lecture Notes in Comput. Sci. 1423, Springer, Berlin, 463–474, 1998.
  • X. Li: Upper bounds on $L$-functions at the edge of the critical strip, Int. Math. Res. Not. IMRN (2010), 727–755.
  • N. Ishii, P. Kaplan and K.S. Williams: On Eisenstein's problem, Acta Arith. 54 (1990), 323–345.
  • I. Niven, H.S. Zuckerman and H.L. Montgomery: An Introduction to the Theory of Numbers, fifth edition, Wiley, New York, 1991.
  • J. Park: Inverse problem for Pell equation and real quadratic fields of the least type, preprint.
  • E.V. Podsypanin: The length of the period of a quadratic irrationality, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 82 (1979), 95–99, (Russian).
  • C. Reiter: Effective lower bounds on large fundamental units of real quadratic fields, Osaka J. Math. 22 (1985), 755–765.
  • G.J. Rieger: Über die Anzahl der als Summe von zwei Quadraten darstellbaren und in einer primen Restklasse gelegenen Zahlen unterhalb einer positiven Schranke, II, J. Reine Angew. Math. 217 (1965), 200–216, (German).
  • T. Takagi: Shoto Seisuron Kogi, Kyoritsu, Tokyo, 1931, (Japanese).
  • Y. Yamamoto: Real quadratic number fields with large fundamental units, Osaka J. Math. 8 (1971), 261–270.