Some New Results About Trigonometry in Finite Fields

Habib Hosseini; Naser Amiri

doi:doi:10.11648/j.pamj.20160504.11

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Some New Results About Trigonometry in Finite Fields

Habib Hosseini, Naser Amiri

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 4)

Received: 23 April 2016 Accepted: 21 May 2016 Published: 17 June 2016

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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 K ⊆ F or there exist a monomorphism f: K → F. 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.

Published in	Pure and Applied Mathematics Journal (Volume 5, Issue 4)
DOI	10.11648/j.pamj.20160504.11
Page(s)	93-96
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Trigonometry, Finite Field, Primitive, Root of Unity

References

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[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.

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Habib Hosseini, Naser Amiri. (2016). Some New Results About Trigonometry in Finite Fields. Pure and Applied Mathematics Journal, 5(4), 93-96. https://doi.org/10.11648/j.pamj.20160504.11

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ACS Style

Habib Hosseini; Naser Amiri. Some New Results About Trigonometry in Finite Fields. Pure Appl. Math. J. 2016, 5(4), 93-96. doi: 10.11648/j.pamj.20160504.11

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Habib Hosseini, Naser Amiri. Some New Results About Trigonometry in Finite Fields. Pure Appl Math J. 2016;5(4):93-96. doi: 10.11648/j.pamj.20160504.11

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@article{10.11648/j.pamj.20160504.11,
  author = {Habib Hosseini and Naser Amiri},
  title = {Some New Results About Trigonometry in Finite Fields},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {4},
  pages = {93-96},
  doi = {10.11648/j.pamj.20160504.11},
  url = {https://doi.org/10.11648/j.pamj.20160504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160504.11},
  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 K ⊆ F or there exist a monomorphism f: K → F. 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.},
 year = {2016}
}

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TY  - JOUR
T1  - Some New Results About Trigonometry in Finite Fields
AU  - Habib Hosseini
AU  - Naser Amiri
Y1  - 2016/06/17
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160504.11
DO  - 10.11648/j.pamj.20160504.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
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PB  - Science Publishing Group
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AB  - 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 K ⊆ F or there exist a monomorphism f: K → F. 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.
VL  - 5
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Author Information

Habib Hosseini

Department of Mathematics, Firoozabad Branch, Islamic Azad University, Firoozabad, Iran
Naser Amiri

Department of Mathematics, Tehran Payame Noor University, Tehran, Iran

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APA Style

Habib Hosseini, Naser Amiri. (2016). Some New Results About Trigonometry in Finite Fields. Pure and Applied Mathematics Journal, 5(4), 93-96. https://doi.org/10.11648/j.pamj.20160504.11

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ACS Style

Habib Hosseini; Naser Amiri. Some New Results About Trigonometry in Finite Fields. Pure Appl. Math. J. 2016, 5(4), 93-96. doi: 10.11648/j.pamj.20160504.11

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AMA Style

Habib Hosseini, Naser Amiri. Some New Results About Trigonometry in Finite Fields. Pure Appl Math J. 2016;5(4):93-96. doi: 10.11648/j.pamj.20160504.11

Copy | Download

@article{10.11648/j.pamj.20160504.11,
  author = {Habib Hosseini and Naser Amiri},
  title = {Some New Results About Trigonometry in Finite Fields},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {4},
  pages = {93-96},
  doi = {10.11648/j.pamj.20160504.11},
  url = {https://doi.org/10.11648/j.pamj.20160504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160504.11},
  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 K ⊆ F or there exist a monomorphism f: K → F. 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.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Some New Results About Trigonometry in Finite Fields
AU  - Habib Hosseini
AU  - Naser Amiri
Y1  - 2016/06/17
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160504.11
DO  - 10.11648/j.pamj.20160504.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 93
EP  - 96
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160504.11
AB  - 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 K ⊆ F or there exist a monomorphism f: K → F. 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.
VL  - 5
IS  - 4
ER  -

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