WELCOME TO THE

PRESENTATION

A SHORT STUDY OF GALOIS FIELD

Course Title: 4th Year Honors Project

Course Number: MTH 490

Presented By

Exam Roll Number: 2011

Reg. Number: H-1468

Admission Session: 2008-2009

OBJECTIVES:

To discuss the preliminaries of the project

Introduction of Galois Field

Examples of Galois Field

To discuss the related theorems of Galois Field

Computational approach of Galois Field

Applications of Galois Field

PRELIMINARIES

Set

Relation

Function

Binary operation

Group and related definitions

Ring and related definitions

Field and characteristic of field

SET:

A set is a well-defined collection of distinct objects. The

objects that make up a set (also known as the elements or

members of a set) can be anything: numbers, people, letters

of the alphabet and so on.

GROUP:

A non-empty set G is said to be a group in G there is defined

an operation “*” such that the following axioms are satisfied:

Closure property

Associative law

Existence of the identity element

Existence of the inverse of each element

GALOIS FIELD

Finite field

Definition of Galois field

example and theorem

Galois field is cyclic

Characteristic of Galois field

FINITE FIELD:

A field having only a finite number of elements is called a

finite field. Simply, a Galois field is a special case of finite

field.

GALOIS FIELD:

Galois Field :

A field in which the number of elements is of the form pn

where p is a prime and n is a positive integer, is called a

Galois field, such a field is denoted by GF (pn).

Example: GF (31) = {0, 1, 2} for ( mod 3) form a finite field of

order 3

DESCRIPTION OF THE EXAMPLE:

For GF-3.The elements are 0, 1 and 2. The multiplication

table is:

Table of reciprocals :

* 1 2

1 1 2

2 2 1

1 2

1 2

THE ADDITION TABLE IS:

The additive identity is 0

The additive inverse of 0 is 0

The additive inverse of 1 is 2

The additive inverse of 2 is 1

Here,GF-3 satisfied all the properties of Galois Field. So GF-3 is a Galois

Field of order 3.

+ 0 1 2

0 0 1 2

1 1 2 0

2 2 0 1

THEOREMS OF GALOIS FIELD:

The multiplicative group of GF ( pn ) is cyclic, Where p is a

prime number and n is an integer.

GF(pn)has a subfield 𝐹′ with pm elements if and only if m|n .

Moreover,𝐹′ is unique.

Let F be a finite field. Then the number of elements of F is pn for

some positive integer n.

THEOREMS OF GALOIS FIELD:

Let F be a finite field with pn elements and let α ∈ F. Then there exist

elements μ and ν in F such that α= μ 2+ ν2

Each element of a finite field with elements satisfies the equation

𝑥𝑝𝑛= x.

COMPUTATIONAL APPROACH OF GALOIS

FIELD

Verification of sum of two squares theorem of last chapter

Some examples of Galois field

Finite field arithmetic (Addition & subtraction)

Primitive Polynomial

Application

VERIFICATION:

Here I have verified the theorem α = μ 2+ ν2 , where α, μ, ν ∈ F and F is a Field of pn elements. I

verified this theorem by FORTRAN programming language.

I verified the theorem for p =11, n = 1, i.e. for Galois field GF (11).

𝑍11 = {0, 1,2,3,4,5,……..9,10 } is a field. Thus we may consider GF(11) = 𝑍11. We can easily check

that every element of 𝑍11 satisfy the polynomial 𝑥11-x 𝑍11 by using FORTRAN programming

language, where every αGF(11) and µ, ν GF(11).

SOLUTION BY FORTRAN:

DIMENSION MAT (100)

INTRGER MAT, K1, CAL, REM

WRITE (*,*)’ENTER A PRIME NUMBER:’

READ (*,*) K1

DO 4 I=K1, 1

MAT (I) =I-1

4 CONTINUE

WRITE (*,*)’REQUIRED ROOTS ARE IN 𝑍𝑃 :’

WRITE (*,*) (MAT (I), I=1, K1, 1)

WRITE (*,*)’

WRITE (*,*)’EVERY ROOT CAN BE EXPRESSED AS:’

DO 1 I=1, K1, 1

DO 2 J=I, K1, 1

DO 3 K=J, K1, 1

CAL= (MAT (J) **2) + (MAT (K) **2)

REM=MOD (CAL, K1)

IF (MAT (I).EQ.REM) THEN

WRITE (6, 5) MAT (I), MAT (J), MAT (K)

5 FORMAT(1X,I2,’=’,I2,’^2+’,I2,’^2’)

GO TO 1

END IF

3 CONTINUE

2 CONTINUE

1 CONTINUE

STOP

END

OUTPUT OF THE PROGRAM:

Input: The Prime Number 11

Output: Required Roots are in 𝑍𝑃 is

0 1 2 3 4 5 6 7 8 9 10

Every Root Can be expressed as:

0 = 02 + 02 1 = 02 + 12 2 = 12 + 12

3 = 02 + 42 4 = 02 + 22 5 = 12 + 22

6 = 32 + 62 7 = 22 + 42 8 = 22 + 22

9 = 02 + 32 10 = 12+ 32

Since it is possible to write α=µ2 +𝜐2 where α, µ,𝜐 𝑍11 then we conclude that every root can be

expressed as the sum of two squares.

REFERENCES:

Hiram Palely and Paul M. Weichsel: “A First Course in Abstract Algebra” New York, Holt, 1996.

R. S. Aggarwal: A text book on modern algebra

Mary Gray: “A radical approach to algebra”, Addison-Wesley publishing Co. London, 1970.

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

Bhattacharya, P.B. adds Jain, S.K., and Naipaul: “A first course in rings, fields and vector spaces,

Halsted Press, New York, 1977.

www.mathworld.wolfarm.com

https://www.wikipedia.org/

http://www.wikihow.com/Main-Page

http://stackoverflow.com/

www.andrew.edu

www.encyclopedia.com

THANK YOU