1. Multiplicative Group The multiplicative group of Z n includes every a, 0<a<n, such that (a,n)=1. The number of elements is Euler’s Totient function  (n) If n is prime,  (n)=n-1 If n=PQ, and P, Q are prime then  (n)=(P-1)(Q-1)

2. Order of Elements Let a n denote a ,…,  a n times We say that a is of order n if a n =1, and for any 0<m<n, a m  1 Examples Euler theorem: in the multiplicative group of Z n any element is of order at most  (n) Generalization: in a finite group every element has finite order and it is at most the size of the group.

3. Sub-groups Let (G,  ) be a group. (H,  ) is a sub-group of (G,  ) if it is a group, and H  G Claim: If (G,  ) is a finite group and (H,  ) is closed, where H  G, then (H,  ) is a sub-group of (G,  ). Examples Lagrange theorem: if G is finite and (H,  ) is a sub-group of (G,  ) then |H| divides |G| Examples

4. Cyclic Groups Claim: let G be a group and a be an element of order n. The set [a]={1, a,…,a n-1 } is a sub-group of G, and is called the sub-group generated by a. a is the generator of [a] If G is generated by some a, G is called cyclic. Theorem: for any prime p, the multiplicative group of Z p is cyclic

5. Rings and Fields

6. Rings A ring has properties as follows: – is a commutative group, identity 0 – is associative, identity 1 –  is distributive: a  (b  c)=(a  b)  (a  c) and (b  c)  a=(b  a)  (c  a) A ring is called commutative if  is a commutative operation Claim: 0 is a multiplicative annihilator in a ring Examples: Z, Z n

7. Fields A field is a commutative ring in which all non-zero elements have multiplicative inverses Example: Rational numbers, Z p Claim: In a field there are no r,s  0, r  s=0 Theorem: the multiplicative group of a finite field is cyclic

8. Polynomials over Rings A polynomial is an expression: a(x)=a m x m  …  a 0 over a commutative ring (where x m denotes x  x) Degree of a polynomial is m, for the largest non-zero a m

9. Polynomial Ring The Polynomial Ring R[x] is a commutative ring over a commutative ring R: –Addition: c(x)=a(x)+b(x) if c i =a i  b i –Multiplication: d(x)=a(x)b(x) if d i =a 0  b i  a 1  b i-1  …  a i  b 0 Examples of operations over Z 2. Is Z 2 [x] finite?

10. Analogies Polynomials-Integers Henceforth we consider polynomials over finite fields If h(x)  0, there is a unique representation for g(x) as g(x)=q(x)h(x)+r(x) such that degree r(x) < degree h(x) Example If for g(x) there is no h(x) of degree>0 s.t. h(x)|g(x) then g(x) is irreducible Example x 4 +x+1 is irreducible over Z 2 (not over Z 3 !)

11. F[x]/f(x) F[x]/f(x): includes all polynomials over field F of degree less than the degree of f(x). Addition and multiplication are computed modulo f(x) F[x]/f(x) is a commutative ring for any f(x) GCD theorems carry over to F[x]/f(x) –Euclidean algorithm finds (g(x),h(x)) –Extended Euclid finds a(x), b(x) s.t. a(x)g(x)+b(x)h(x)=(g(x),h(x)) Theorem: If f(x) is irreducible F[x]/f(x) is a field

12. Finite Fields The characteristic of a field F is the smallest m such that 1  1 m times is 0 Claim: In a finite field the characteristic is prime Example: The characteristic of Z p is p Theorem: All finite fields with the same number of elements are identical up to isomorphism Theorem: the number of elements of a finite field is p n for a prime p and a natural number n. Example: GF(2 4 ) with irreducible poly. x 4 +x+1