International Journal of Advanced and Applied Sciences

Int. j. adv. appl. sci.

EISSN: 2313-3724

Print ISSN: 2313-626X

Volume 4, Issue 2  (February 2017), Pages:  22-27


Title: A computational technique for determining the fundamental unit in explicit types of real quadratic number fields

Author(s):  Özen Özer 1, *, Abdel Badeh M. Salem 2

Affiliation(s):

1Department of Mathematics, Faculty of Science and Arts, Kırklareli University, 39100, Kırklareli, Turkey
2Faculty of Computer and Informatic Science, Ain Shams University, Cairo, Egypt

https://doi.org/10.21833/ijaas.2017.02.004

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Abstract:

In real quadratic number field Q(Sqrt (d)), integral basis element is denoted by  wd= [a0; a1 to al] for the period length   l(d). The fundamental unit  of real quadratic number field is also denoted by Epsilond. The Unit Theorem for real quadratic fields says that every unit in the integer ring of a quadratic field is generated by the fundamental unit. Also, regulator in real quadratic cryptography is outstanding.  We have seen that the regulator  R plays the role of a group order. The regulator problem is to find an integer   R' satisfies |R'-R|<1   where   R' is an approximation of  with any given precision can be computed in polynomial time for discriminant. However, some of the fundamental units can not be calculated by computer programme in short time because of the big numbers  or long calculations of usual algorithm. This is also the main problem from the computing/informatics point of view. So, determining of the fundamental units is of great importance. In this paper, we construct a theorem to determine the some certain real quadratic fields Q(Sqrt(d)) having specific form of continued fraction expansion of wd where d is a square-free integer. We also present the general context and obtain new certain parametric representation of fundamental unit for such types of fields.By specialization, we get a fix on Yokoi’s invariants and support all results with tables. 

© 2017 The Authors. Published by IASE.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Quadratic fields, Continued fractions, Fundamental units, Yokoi’s invariants

Article History: Received 24 October 2016, Received in revised form 10 January 2017, Accepted 10 January 2017

Digital Object Identifier: 

https://doi.org/10.21833/ijaas.2017.02.004

Citation:

Özer Ö and Salem ABM (2017). A computational technique for determining the fundamental unit in explicit types of real quadratic number fields. International Journal of Advanced and Applied Sciences, 4(2): 22-27

http://www.science-gate.com/IJAAS/V4I2/Özer.html


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