Download (a,n)=1

Multiplicative Group
•
•
•
•
The multiplicative group of Zn 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)
Order of Elements
• Let an denote a,…,a n times
• We say that a is of order n if an=1, and for any
0<m<n, am1
• Examples
• Euler theorem: in the multiplicative group of Zn
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.
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
Cyclic Groups
• Claim: let G be a group and a be an element of
order n. The set [a]={1, a,…,an-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 Zp is cyclic
Rings and Fields
Rings
• A ring <R,,> has properties as follows:
– <R,> is a commutative group, identity 0
– <R,> 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, Zn
Fields
• A field is a commutative ring in which all
non-zero elements have multiplicative
inverses
• Example: Rational numbers, Zp
• Claim: In a field there are no r,s0, rs=0
• Theorem: the multiplicative group of a
finite field is cyclic
Polynomials over Rings
• A polynomial is an expression:
a(x)=amxm…a0 over a commutative ring
<R,,> (where xm denotes xx)
• Degree of a polynomial is m, for the largest
non-zero am
Polynomial Ring
• The Polynomial Ring R[x] is a commutative
ring over a commutative ring R:
– Addition: c(x)=a(x)+b(x) if ci=aibi
– Multiplication: d(x)=a(x)b(x) if
di=a0bia1  bi-1  …  ai  b0
• Examples of operations over Z2. Is Z2[x]
finite?
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 x4+x+1 is irreducible over Z2(not over
Z3!)
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
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 Zp 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 pn for a prime p and a natural number n.
• Example: GF(24) with irreducible poly. x4+x+1