M. Khalily Dermany Islamic Azad University

 finite number of element  important in number theory, algebraic geometry, Galois theory, cryptography, coding theory and Quantum error correction  applications in many areas of mathematics and computer science, including coding theory, LFSRs, modular representation theory, and the groups of Lie type  Finite fields are an active area of research, including recent results on the Kakeya conjecture and open problems on the size of the smallest primitive root.

 Semi group ◦ Associative: (x+y)+z=x+(y+z)  Monoid ◦ A semi group with identity: a + e = a  Group ◦ A Monoid with inverses : a + (−a) = e ◦ The order of a group is the number of elements in the group.  Abelian group ◦ Commutativity: a + b = b + a  Ring ◦ is Abelian and is group ◦ Distributivity: a · (b + c) = (a · b) + (a · c).  Field ◦ and is Abelian ◦ Distributivity a · (b + c) = (a · b) + (a · c).

 Closure of F under addition and multiplication ◦ For all a, b in F, both a + b and a · b are in F (or more formally, + and · are binary operations on F).  Associativity of addition and multiplication ◦ For all a, b, and c in F, a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.  Commutativity of addition and multiplication ◦ For all a and b in F : a + b = b + a and a · b = b · a.

 Existence of additive and multiplicative identity elements ◦ There exists an element of F, called the additive identity element and denoted by 0, such that for all a in F  a + 0 = a ◦ Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F  a · 1 = a  the additive identity and the multiplicative identity are required to be distinct.

 Existence of additive inverses and multiplicative inverses or subtraction and division operations exist. ◦ For every a in F, there exists an element −a in F, such that a + (−a) = 0 ◦ Similarly, for any a in F other than 0, there exists an element a −1 in F, such that a · a −1 = 1. ◦ The elements a + (−b) and a · b −1 are also denoted a − b and a/b, respectively  Distributivity of multiplication over addition ◦ For all a, b and c in F, the following equality holds: a · (b + c) = (a · b) + (a · c)

 example F 4 is a field with four elements  Inverse  Identity

 all finite fields must have prime power order ◦ there is no finite field with 6 elements.  In any field F with m elements, the equation x m =x is satisfied by all elements x of F.  In any prime size field, it can be proved that there is always at least one element whose powers constitute all the nonzero elements of the field. This element is said to be primitive.

 For example, in the field GF(7), the number 3 is primitive as  6 x 2=3 3 x 3 2 = 3 5 = 5

 subtract 6 from 3, ◦ first use the addition table to find the additive inverse of 6, which is 1. ◦ Then we add 1 to 3 to obtain the result ◦ 3-6=3+(-6)=3+1=4  divide 3 by 2. ◦ first find the multiplicative inverse of 2, which is 4, ◦ multiply 3 by 4 to obtain the result ◦ 3÷2=3.(2 -1 )=3.4=5.

 polynomials whose coefficients are from the binary field GF(2)  The degree of a polynomial is the largest power of X with a nonzero coefficient.

 There are two polynomials over GF(2) with  degree 1 ◦ X and 1+X.  There are four polynomials over GF(2) with  degree 2 ◦ X 2, 1 + X 2, X + X 2, and 1 + X + X 2  In general, there are 2 n polynomials over GF(2) with degree n.

 Therefore, the set F* is a Galois field of 2 m elements. Also GF(2) is a subfield of GF(2 m ).

 X 2 + 6X + 25 does not have roots ◦ —3 + 4j ◦ —3 — 4j  This is also true for polynomials with coefficients from GF(2)

 there must exist two positive integers m and n such that m < n and  There must exist a smallest positive integer λ such that λ  This integer λ is called the characteristic of the field GF(q).  λ is a prime.

 Any two finite fields with the same number of elements are isomorphic. That is, under some renaming of the elements of one of these, both its addition and multiplication tables become identical to the corresponding tables of the other one.