Visualizing Finite Field of Order P 2 Through Matrices

Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 16 Issue 1 Version 1.0 Year 2016 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249 -4626 & Print ISSN: 0975-5896

Visualizing Finite Field of Order p2 through Matrices By S. K. Pandey Sardar Patel University of Police, India Abstract- In this article we consider finite field of order p2 (p ≠ 2). We provide matrix representation of finite field of order at most p2 = 121. However this article provides a technique to yield matrix field of order p2 for every prime p > 2 . Keywords: finite field, matrix field, Galois field. GJSFR-F Classification : MSC 2010: 12E20

VisualizingFiniteFieldofOrderp2throughMatrices

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Ref Visualizing Finite Field of Order p2

through Matrices 201 r ea Y S. K. Pandey 271

Abstract- In this article we consider finite field of order 2 (pp ≠ 2). We provide matrix representation of finite field of order at most p2 =121 . However this article provides a technique to yield matrix field of order p2 for every prime p > 2 . V I

Keywords: finite field, matrix field, Galois field. ue ersion I s s

MSCS2010: 12E20. I

I. Intro duction XVI There is a finite field of order pn for every positive prime p and positive integer n . If is a finite field of order and is the polynomial domain defined over then one F p []xF F

) F

F[]x ( can construct a finite field K = of order pn , where f ()x ∈ F[]x is an irreducible [] ()xf

polynomial of degree n and (xf ) ][ is the ideal generated by (xf ). This field K has a

subfield isomorphic with the field F . One may refer [1, 2] for further details. This is the Research Volume usual method of construction of finite fields. In this article we construct finite matrix fields of order 2 without adopting the p Frontier usual method of construction of finite fields. We give finite matrix field of order at most 121. However following the approach given in this article one can construct finite Science matrix field of order p2 for each p ≠ .2 of 2 II. Finite Matrix Field of Order p

In [3] one can find different matrix representations of a finite field of order p . Let Journal

p be a prime number. Then Z p = { ,4,3,2,1,0 5 p −1... }is a field under addition and Global Pandey, Matrix ofField Finite and Infinite Order, Research of Journal Pure . multiplication modulo p . One matrix representation of Z p as given in [3] is a 0  S. K S. Algebra (accepted). (accepted).Algebra   Fp =   : a ∈ Z p . Every field of characteristic p contains a copy of a field of order 0 a 3.  

Author: Department of Mathematics, Sardar Patel University of Police, Security and Criminal Justice Daijar, Jodhpur, Rajasthan., India. e mail: skpandey12@@gmail.com

p . All the fields given below are finite matrix field of order p 2 . One can easily see that

all these fields contain Fp . We use very simple technique to obtain matrix fields of order p2 . It is seen that  ba   F =   a,: ∈ Rb is isomorphic to the field Cof complex numbers. Here R stands − b a  for the field of real numbers. We notice that if we replace R by Z then we obtain a p finite matrix field of order p 2 . It may be noted that this approach does not work for Notes p = 2 .

201 a) Finite Matrix Field Of Order9 r

Yea  00   01   02   10   20   11   21   12  2 2 =                   28 M 9  , , , , , , , ,   00   10   20   02   01   12   11  2 2  21 

This is a finite field of order9under addition and multiplication of matrices   V  00   01  2 0 modulo3 . It is a field of characteristic 3 and it contains F =  , ,  . I 3        00  0 1  20  ue ersion I s s

I b) Finite Matrix Field Of Order 25

XVI  00   01   02   03   04   11   41   12   42   43   13    , , , , , , , , , , ,   00   10   20   30   40   14   11   24   21   31   34    14   44   40   10   21   31   30   20   22   32   33   =                       ) M 25  , , , , , , , , , , ,

F  44   41   01   04   13   12   02   03   23   22   32   (  23   24  4 3   , ,    33  3 4  42  

Research Volume This is a finite field of order 25 under addition and multiplication of matrices modulo5 . One can see that it is a field of characteristic 5and it contains Frontier  00   01   02   03  4 0           . F5 =  , , , ,   00   10   20  0 3  40  Science c) Finite Matrix Field Of Order 49 of  00   01   02   03   04   05   06   60   10   20   50    , , , , , , , , , , , Journal  00   10   20   30   40   50   60   01   06   05   02    40   30   11   61   21   51   41   31   32   42   52    , , , , , , , , , , , 

Global  03   04   16   11   15   12   13   14   24   23   22      22   62   12   43   33   53   23   13   63   44   34                         M 49 =  , , , , , , , , , , ,   25   21   26   33   34   32   35   36   31   43   44    54   24   14   64   45   35   55   25   65   15   46    , , , , , , , , , , ,  42   45   46   41   53   54   52   55   51   56   63      36   56   26   16  6 6             , , , ,    64   62   65  6 6  61  

This is a finite field of order 49 under addition and multiplication of matrices modulo 7 . One can see that it is a field of characteristic 7 and it contains

 00   01   02   03  4 0  05   06                F7 =  , , , , , ,  .  00   10   20   30  0 4  50  0 6

d) Finite Matrix Field Of Order121 Notes  00   01   02   03   04   05   06   07   08   09   010    , , , , , , , , , , ,  00   10   20   30   40   50   60   70   80   90   100     201  100   10   90   20   80   30   40   70   60   50   101  r

 , , , , , , , , , , , ea

                      Y  01   010   02   09   03   08   07   04   05   06   11    291  11   21   91   81   31   41   71   61   51   102   12    , , , , , , , , , , ,  110   19   12   13   18   17   14   15   16   21   210      92  22   82   32   72   42   62   52   103   13   93     , , , , , , , , , ,  V  22  29   23   28   24   27   25   26   31   310   32   I ue ersion I

  s

 23   83   33   73   43   63   53   104   14   24   94  s  , , , , , , , , , , ,                        I  39   33   38   34   37   35   36   41   410   49   42     XVI  84   34   74   44   64   54   105   15   95   25   35    , , , , , , , , , , ,  43   48   44   47   45   46   51   510   52   59   58   M121= 

 85   75   45   65   55   106   16   96   26   86   36  )                         , , , , , , , , , , , F  53 54 57 55 56 61 610 62 69 63 68  (                         76   46   66   56   107   17   97   27   87   37   77   , , , , , , , , , , ,  64   67   65   66   71   710   72   79   73   78   74                        Research Volume  47   67   57   108   18   98   28   88   38   78   48    , , , , , , , , , , ,   77 75 76 81 810 82 89 83 88 84 87 

                      Frontier    68   58   109   19   99   29   89   39   79   49   69   , , , , , , , , , , ,   85   86   91   910   92   99   93   98   94   97   95  

                      Science  

 59   1010   110   910   210   810   310   410   710   of  , , , , , , , , ,   96   101   1010   102   109   103   108   107   104  

 610  10 5   Journal  ,         5 10  106   Global This is a finite field of order 121 under addition and multiplication of matrices modulo11. It has characteristic 11 and it contains

 00   01   02   03   04   05   06   07   08   09   010                        F11 =  , , , , , , , , , ,   00   10   20   30   40   50   60   70   80   90   100 

Following same technique one can find finite matrix field of order p 2 for p > .11

References Références Referencias

1. M. Artin, Algebra, Prentice Hall of India Private Limited, New Delhi, 2000. 2. T. W. Hungerford, Algebra, Springer-India, New Delhi 2005. 3. S. K. Pandey, Matrix Field of Finite and Infinite Order, Research Journal of Pure Algebra 5 (12), 214-216, 2015.

Notes

201

Yea 30 V I ue ersion I s s I XVI ) F ( Research Volume Frontier Science of Journal Global