Galois field

This is my honors project presentation for "A short study of Galois Field"

1. WELCOME TO THE PRESENTATION

2. 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

3. 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

4. PRELIMINARIES Set Relation Function Binary operation Group and related definitions Ring and related definitions Field and characteristic of field

5. 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.

6. 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

7. GALOIS FIELD Finite field Definition of Galois field example and theorem Galois field is cyclic Characteristic of Galois field

8. 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.

9. GALOIS FIELD: Galois Field : A field in which the number of elements is of the form p n where p is a prime and n is a positive integer, is called a Galois field, such a field is denoted by GF ( p n ). Example: GF (3 1 ) = {0, 1, 2} for ( mod 3) form a finite field of order 3

10. 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

11. 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. + 1 2 1 2 1 1 2 2 2 1

12. THEOREMS OF GALOIS FIELD: The multiplicative group of GF ( p n ) is cyclic, Where p is a prime number and n is an integer . GF( p n )has a subfield with p m elements if and only if m|n . Moreover, is unique . Let F be a finite field. Then the number of elements of F is p n for some positive integer n .  

13. THEOREMS OF GALOIS FIELD: Let F be a finite field with p n elements and let α F. Then there exist elements μ and ν in F such that α= μ 2 + ν 2 Each element of a finite field with elements satisfies the equation = x.  

14. 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

15. VERIFICATION: Here I have verified the theorem α = μ 2 + ν 2 , where α, μ, ν F and F is a Field of p n 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). = , , , , ,…….. , } is a field. Thus we may consider GF(11) = . We can easily check that every element of satisfy the polynomial - x  by using FORTRAN programming language, where every α  GF(11) and µ , ν  GF(11).  

16. 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 :’ WRITE (*,*) (MAT (I), I=1, K1, 1) WRITE (*,*)’ WRITE (*,*)’EVERY ROOT CAN BE EXPRESSED AS:’  

17. 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

18. OUTPUT OF THE PROGRAM: Input: The Prime Number 11 Output: Required Roots are in is 0 1 2 3 4 5 6 7 8 9 10 Every Root Can be expressed as: 0 = 0 2 + 2 1 = 0 2 + 1 2 2 = 1 2 + 1 2 3 = 0 2 + 4 2 4 = 0 2 + 2 2 5 = 1 2 + 2 2 6 = 3 2 + 6 2 7 = 2 2 + 4 2 8 = 2 2 + 2 2 9 = 2 + 3 2 10 = 1 2 + 3 2 Since it is possible to write α= where α, µ ,  then we conclude that every root can be expressed as the sum of two squares.  

19. 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

20. THANK YOU