Volume 5, Issue 4, August 2016, Page: 93-96
Some New Results About Trigonometry in Finite Fields
Habib Hosseini, Department of Mathematics, Firoozabad Branch, Islamic Azad University, Firoozabad, Iran
Naser Amiri, Department of Mathematics, Tehran Payame Noor University, Tehran, Iran
Received: Apr. 23, 2016;       Accepted: May 21, 2016;       Published: Jun. 17, 2016
DOI: 10.11648/j.pamj.20160504.11      View  3071      Downloads  124
Abstract
In this paper we study about trigonometry in finite field, we know that , the field with p elements, where p is a prime number if and only if p = 8k + 1 or p = 8k + 1 or p = 8k + 1 or p = 8k−1. Let F and K are two field, we say that F is an extension of K, if KF or there exist a monomorphism f: KF. recall that , F[x] is the ring of polynomial over F. If K F (means that F is an extension of K) an element u εF is algebraic over K if there exists f(x) ε K[x] such that f(u)=0. The algebraic closure of K in F is , is the set of all algebraic elements in F over K.
Keywords
Trigonometry, Finite Field, Primitive, Root of Unity
To cite this article
Habib Hosseini, Naser Amiri, Some New Results About Trigonometry in Finite Fields, Pure and Applied Mathematics Journal. Vol. 5, No. 4, 2016, pp. 93-96. doi: 10.11648/j.pamj.20160504.11
Copyright
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
Carl stitz, Jeff zeager, College Trigonometry, lorain county community college, july 4, 2013.
[2]
R. M. Campello de Souza, H. M. de Oliveira and D. Silva, "The Z Transform over Finite Fields," International Telecommunications Symposium, Natal, Brazil, 2002.
[3]
M. M. Campello de Souza, H. M. de Oliveira, R. M. Campello de Souza and M. M. Vasconcelos, "The Discrete Cosine Transform over Prime Finite Fields," Lecture Notes in Computer Science, LNCS 3124, pp. 482–487, Springer Verlag, 2004.
[4]
R. M. Campello de Souza, H. M. de Oliveira and A. N. Kauffman, "Trigonometry in Finite Fields and a New Hartley Transform," Proceedings of the 1998 International Symposium on Information Theory, p. 293, Cambridge, MA, Aug. 1998.
[5]
F. Delbaen, W. Schachermayer, The Mathematics of Arbitrage, Springer, Berlin, 2006.
[6]
Chowla, S, The Riemann hypottesis Hilbert’s tenth problem, Gorden and Breach Seience Publishers (1965).
[7]
Lang, S, Algebra, second Edition Addison - Wesley publishing company 1994.
[8]
K. Gustafson, Operator trigonometry, Linear and Multilinear Algebra, 37:139-159 (1994).
[9]
K. Gustafson, Operator trigonometry of iterative methods, Numer. Linear Algebra Appl. 4 (1997) 333–347.
[10]
J. B. Lima, R. M. Campello de Souza, Fractional cosine and sine transforms over finite fields, Linear Algebra and its Applications, Volume 438, Issue 8, 15 April 2013, Pages 3217-3230.
[11]
Karl Gustafson, Operator trigonometry of multivariate finance, Journal of Multivariate Analysis, Volume 101, Issue 2, February 2010, Pages 374-384.
[12]
John Bird, Geometry and Trigonometry, Engineering Mathematics Pocket Book (Fourth Edition), 2008, Pages 105-148.
[13]
Honghai Liu, George M. Coghill, Dave P. Barnes, Fuzzy qualitative trigonometry, International Journal of Approximate Reasoning, Volume 51, Issue 1, December 2009, Pages 71-88.
[14]
Gauss, C. F., Disquisitrones arithmetica, Bravncshweig 1801, English translation, Yale paperbound, 1965.
[15]
Rudolf Lidl, Harald Niederreiter, Finite Fields and Their Applications, second edition, Cambridge University Press, 1994.
Browse journals by subject