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TO THE

PRESENTATION

Presented By

Exam Roll: 1820

Registration No: 2010-712-250

Department Of Mathematics

University Of Dhaka

A SHORT STUDY OF GALOIS FIELD

Objectives Of The Project

To introduce Galois Field.

To discuss related theorems.

Computational approach of Galois Field.

Applications of Galois Field.

What is Galois field ?

A Galois field(so-named in honor of Évariste Galois) is a

field in which the number of elements is always a

positive integer power of a prime number.

That is, the number of elements of a

Galois field is of the form pn , where p is

a prime and n is a positive integer.

Generally it is denoted by GF(pn ).

Évariste Galois(1811-1832)

Examples

1. GF(31)={0,1,2} forms a Galois field.

Addition table

Where 0 is the additive identity element and

0 is the additive inverse of 0

1 is the additive inverse of 2

2 is the additive inverse of 1

Addition 0 1 2

0 0 1 2

1 1 2 0

2 2 0 1

Multiplication 1 2

1 1 2

2 2 1

Multiplication Table

Where 1 is the multiplicative identity element and

1 is the multiplicative inverse of 1

2 is the multiplicative inverse of 2

2. GF-13:The elements are 0,1,2,3,4,5,6,7,8,9,a,b and c.

Here a=10, b=11, c =12.

Addition Table

Additi

on

0 1 2 3 4 5 6 7 8 9 a b c

0 0 1 2 3 4 5 6 7 8 9 a b c

1 1 2 3 4 5 6 7 8 9 a b c 0

2 2 3 4 5 6 7 8 9 a b c 0 1

3 3 4 5 6 7 8 9 a b c 0 1 2

4 4 5 6 7 8 9 a b c 0 1 2 3

5 5 6 7 8 9 a b c 0 1 2 3 4

6 6 7 8 9 a b c 0 1 2 3 4 5

7 7 8 9 a b c 0 1 2 3 4 5 6

8 8 9 a b c 0 1 2 3 4 5 6 7

9 9 a b c 0 1 2 3 4 5 6 7 8

a a b c 0 1 2 3 4 5 6 7 8 9

b b c 0 1 2 3 4 5 6 7 8 9 a

c c 0 1 2 3 4 5 6 7 8 9 a b

Multipl

ication

1 2 3 4 5 6 7 8 9 a b c

1 1 2 3 4 5 6 7 8 9 a b c

2 2 4 6 8 a c 1 3 5 7 9 b

3 3 6 9 c 2 5 8 b 1 4 7 a

4 4 8 c 3 7 b 2 6 a 1 5 9

5 5 a 2 7 c 4 9 1 6 b 3 8

6 6 c 5 b 4 a 3 9 2 8 1 7

7 7 1 8 2 9 3 a 4 b 5 c 6

8 8 3 b 6 1 9 4 c 7 2 a 5

9 9 5 1 a 6 2 b 7 3 c 8 4

a a 7 4 1 b 8 5 2 c 9 6 3

b b 9 7 5 3 1 c a 8 6 4 2

c c b a 9 8 7 6 5 4 3 2 1

Multiplication Table

THEOREMS ON GALOIS FIELD

Theorem 1: A multiplicative group of GF(Pn) is cyclic.

Theorem 2: Let F be a finite field with Pn elements and let

F. Then there exists elements and in F such that

= 2+ 2.

Theorem 3: GF( pn) has a subfield F′ with pm elements if

and only if m | n. Moreover, F' is unique.

Theorem 4: The number of elements in a finite field of

characteristic p is of the form Pn, n being some positive

integer.

Theorem 5: Each element of a finite field with p elements

satisfies the equation 𝑥𝑝𝑛− 𝑥 = 0.

Theorem 6: lf a finite field F has pn elements , then F is

the decomposition field of the polynomial 𝑥𝑝𝑛− 𝑥.

Computational approach of Galois Field

CHECKING

Now we illustrate the example GF-17 by a programming

language MATHEMATICA 5.2.We know that

𝑍17={0, 1,2,3,4,5,…….14, 15, 16 } is a field. By using this

program we can check any GF-n, where n is any prime

number.

Here we start with EXIT to avoid the disturbance of those

symbols which was used in previous time and to show all

the input and output of every line.

Mathematica program for Addition Table

In[1]:= Exit

In[2]:= n=17

Out[2]= 17

In[3]:= v=Table[i,{i,0,n-1}]

Out[3]= {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}

In[4]:= For[i=1,i<= n,i++,

For[j=1,j<= n,j++,

p=v[[i]]+v[[j]];

If[p>= n,r[j]=p-n,r[j]=p]];

q[i]=Table[r[k],{k,1,n}]];

rslt=Table[q[l],{l,1,n}];

In[5]:= TableForm[rslt,TableHeadings->

{{"0","1","2","3","4","5","6","7","8","9","10","11","12","13

","14","15","16"},{"0","1","2","3","4","5","6","7","8","9","

10","11","12","13","14","15","16"}},TableSpacing->{1,1}]

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0

2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1

3 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2

4 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3

5 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4

6 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5

7 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6

8 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7

9 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8

10 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9

11 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10

12 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11

13 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12

14 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13

15 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

16 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Out[5]=

Mathematica program for Multiplication Table

In[1]:= Exit

In[1]:= n=17

In[2]:= 17

In[3]:= v=Table[i,{i,0,n-1}]

In[3]:= {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}

In[4]:= For[i=2,i<=n,i++,

For[j=2,j<=n,j++,

p=v[[i]]*v[[j]];

If[p>n,r[j]=Mod[p,n],r[j]=p]];

q[i]=Table[r[k],{k,2,n}]];

rslt=Table[q[l],{l,2,n}];

In[5]:= TableForm[rslt,TableHeadings-

>Automatic,TableSpacing->{1,1}]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2 2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15

3 3 6 9 12 15 1 4 7 10 13 16 2 5 8 11 14

4 4 8 12 16 3 7 11 15 2 6 10 14 1 5 9 13

5 5 10 15 3 8 13 1 6 11 16 4 9 14 2 7 12

6 6 12 1 7 13 2 8 14 3 9 15 4 10 16 5 11

7 7 14 4 11 1 8 15 5 12 2 9 16 6 13 3 10

8 8 16 7 15 6 14 5 13 4 12 3 11 2 10 1 9

9 9 1 10 2 11 3 12 4 13 5 14 6 15 7 16 8

10 10 3 13 6 16 9 2 12 5 15 8 1 11 4 14 7

11 11 5 16 10 4 15 9 3 14 8 2 13 7 1 12 6

12 12 7 2 14 9 4 16 11 6 1 13 8 3 15 10 5

13 13 9 5 1 14 10 6 2 15 11 7 3 16 12 8 4

14 14 11 8 5 2 16 13 10 7 4 1 15 12 9 6 3

15 15 13 11 9 7 5 3 1 16 14 12 10 8 6 4 2

16 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Out[5]:=

VERIFICATION:

We verify the theorem 2 for p=23, n=1, i.e. for Galois field

GF(23).

We know that 𝑍23={0, 1,2,3,4,5,…….22} is a field. Thus we

may consider GF(23)=𝑍23.

Now using the following computer program, we verify that

for every αGF(23),there exists µ, GF(23) satisfying

α=µ2+𝜐2.Here we use MATHEMATICA 5.2 programming

tools.

In[1]:= Exit

In[1]:= n=23;

In[2]:= v=Table[i,{i,0,n-1}];

In[3]:= For[i=1,i<= n,i++,

For[j=0,j<= n,j++,

For[k=0,k<= n,k++,

mat=v[[j]]^2+v[[k]]^2;

rem=Mod[mat,n];

If[v[[i]]==Šrem,

p[i]=v[[i]];q[i]=v[[j]];r[i]=v[[k]]]]]

Print[p[i],"=",q[i],"^2 +",r[i],"^2"]]

Out[3]:=

0 = 0 ^2 + 0 ^2

1 = 22 ^2 + 0 ^2

2 = 22 ^2 + 22 ^2

3 = 22 ^2 + 18 ^2

4 = 22 ^2 + 16 ^2

5 = 22 ^2 + 21 ^2

6 = 21 ^2 + 18 ^2

7 = 22 ^2 + 12 ^2

8 = 21 ^2 + 21 ^2

9 = 22 ^2 + 13 ^2

10 = 22 ^2 + 20 ^2

11 = 20 ^2 + 18 ^2

12 = 21 ^2 + 13 ^2

13 = 22 ^2 + 14 ^2

14 = 22 ^2 + 17 ^2

15 = 20 ^2 + 12 ^2

16 = 21 ^2 + 14 ^2

17 = 22 ^2 + 19 ^2

18 = 20 ^2 + 20 ^2

19 = 22 ^2 + 15 ^2

20 = 21 ^2 + 19 ^2

21 = 20 ^2 + 14 ^2

22 = 21 ^2 + 15 ^2

Since it is possible to write α=µ2 +𝜐2 where α, µ, 𝑍23then we conclude that every root can be expressed as the

sum of two squares.

APPLICATIONS OF GALOIS FIELD

1. Galois field is used in Cryptography

2. Galois field is used in Coding theory and Combinatorial

design and Quantum error correction.

3. The most commonly used Galois Field is GF(256) i.e

GF(28).

4. All CD and DVD players use computation in Galois

Field, as do many disk storage system.

5. Galois Field is used to develop some Mathematical

Theories , which have a lot of real life application.

REFERENCES[1] Hiram Palely and Paul M. Weichsel: “A First Course in Abstract Algebra”

New York, Holt, 1996.

[2] J.S. Milne:,”Fields and Galois Theory”, The photograph is of Sabre Peak,

Moraine Creek, New Zealand

[3] R. S. Aggarwal: A text book on modern algebra.

[4] Mary Gray: “A radical approach to algebra”, Addison-Wesley publishing

Co. London,1970.

[5] Professor Abdur Rahman : “ Abstract Algebra”,Dhaka,1995.

[6] Bhattacharya, P.B. adds Jain, S.K., and Naipaul: “A first course in rings,

fields and vector spaces, Halsted Press, New York, 1977.

[7] John.B Fraleigh , “A First Course in Abstract Algebra”. Copyright by

Pearson Education, Inc

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