A Study on the Fundamental Unit of Certain Real Quadratic Number Fields

Özen ÖZER^1,

¹Department of Mathematics, Faculty of Science and Arts, Kırklareli Üniversity, Kırklareli

Cite this paper:
Özen ÖZER. A Study on the Fundamental Unit of Certain Real Quadratic Number Fields. Turkish Journal of Analysis and Number Theory. 2018; 6(1):1-8. doi: 10.12691/tjant-6-1-1

Abstract

In this paper, we consider the certain types of real quadratic fields where is a square free positive integer. We obtain new parametric representation of the fundamental unit for such types of fields. Also, we get a fix on Yokoi’s invariants as well as class numbers and support all results with tables.

Keywords:
quadratic fields continued fraction expansions class numbers fundamental units Yokoi’s invariants

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	Benamar H., Chandoul A. and Mkaouar M. (2015). On the Continued Fraction Expansion of Fixed Period in Finite Fields, Canad. Math. Bull. 58, 704-712.
[2]	Clemens L. E., Merill K. D., Roeder D. W. (1995). Continues fractions and series, J. Number Theory 54, 309-317.
[3]	Elezovi´c N.(1997). A note on continued fractions of quadratic irrationals, Math. Commun. 2, 27-33.
[4]	Halter-Koch F.,(1991). Continued fractions of given symmetric period. Fibonacci Quart., 29(4), 298-303.
[5]	Kawamoto F. and Tomita K. (2008). Continued fraction and certain real quadratic fields of minimal type, J.Math.Soc. Japan, 60, 865-903.
[6]	Louboutin S. (1988). Continued Fraction and Real Quadratic Fields, J.Number Theory, 30, 167-176, 1988.
[7]	Mollin R. A. (1996). Quadratics, CRC Press, Boca Rato, FL, 399p.
[8]	Mollin R. A., Williams, H.C. (1992). On Real Quadratic Fields of Class Number Two, Math. of Comp. 59(200), 625-632.
[9]	Olds C. D. (1963). Continued Functions, New York, Random House, 170 p.
[10]	Özer Ö. (2016). On Real Quadratic Number Fields Related With Specific Type of Continued Fractions, Journal of Analysis and Number Theory, 4(2), 85-90.
[11]	Özer Ö. (2016). Notes On Especial Continued Fraction Expansions and Real Quadratic Number Fields, Kirklareli University Journal of Engineering and Science, 2(1), 74-89.
[12]	Perron O. (1950). Die Lehre von den Kettenbrichen, New York: Chelsea, Reprint from Teubner Leipzig, 200 p.
[13]	Sasaki R. (1986). A characterization of certain real quadratic fields, Proc. Japan Acad., 62, Ser. A, No. 3, 97-100.
[14]	Sierpinski W. (1964). Elementary Theory of Numbers, Warsaw: Monografi Matematyczne.
[15]	Tomita, K., 1995. Explicit representation of fundamental units of some quadratic fields, Proceeding Japan Academia, 71, Ser. A, No. 2, 41-43.
[16]	Tomita, K. and Yamamuro K., 2002. Lower bounds for fundamental units of real quadratic fields, Nagoya Mathematical Journal,166, 29-37.
[17]	Williams, K. S., and Buck, N., 1994. Comparison of the lengths of the continued fractions of and Proceeding American Mathematical Society, 120(4), 995-1002, 1994.
[18]	Yokoi H. (1990). The fundamental unit and class number one problem of real quadratic fields with prime discriminant, Nagoya Math. J., 120, 51-59.
[19]	Yokoi H. (1991). The fundamental unit and bounds for class numbers of real quadratic fields, Nagoya Math. J., 124, 181-197.
[20]	Yokoi H., 1993. A note on class number one problem for real quadratic fields. Proc. Japan Acad., 69, Ser. A, 22-26.
[21]	Yokoi H, 1993. New invariants and class number problem in real quadratic fields. Nagoya Math. J., 132, 175-197.
[22]	Zhang, Z. and Yue, Q., 2014. Fundamental units of real quadratic fields of odd class number. Journal of Number Theory 137, 122-129.