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
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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:
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Closure property
Associative law
Existence of the identity element
Existence of the inverse of each element
GALOIS FIELD
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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:
*
1
2
1
1
2
2
2
1
Table of reciprocals :
1
2
1
2
THE ADDITION TABLE IS:
+
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
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.
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
𝑛
𝑝
𝑥 = x.
COMPUTATIONAL APPROACH OF GALOIS
FIELD
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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 𝑥 11 -x  𝑍11 by using FORTRAN programming
language, where every αGF(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
Every Root Can be expressed as:
0
=
02 + 02
1
3
=
02 + 42
4
6
=
32 + 62
7
9
=
02 + 32
10
=
=
=
=
02 + 12
02 + 22
22 + 42
12 + 32
7
8
9
10
2
5
8
=
=
=
12 + 1 2
12 + 2 2
22 + 2 2
Since it is possible to write α=µ2 +𝜐 2 where α, µ,𝜐  𝑍11 then we conclude that every root can be
expressed as the sum of two squares.
REFERENCES:
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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