Galois field

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
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
𝑥 𝑝 𝑛
= 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 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 𝑍 𝑃 :’
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 𝑍 𝑃 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

Galois field

More Related Content

What's hot (20)

Viewers also liked (9)

Similar to Galois field (20)

Recently uploaded (20)

Galois field