Download Chapter 2 Introduction to Finite Field

Chapter 2
Introduction to Finite Field
Lecture 7, February 1, 2011
Recall:
Definition (Ring). A commutative ring (R, +, ·) is a non-empty set R together with
two binary operations: addition (+) and multiplication (·) such that:
1). (R, +) is an abelian group,
2). (R, ·) is associative and commutative, i.e., x · (y · z) = (x · y) · z and xy = yx for all
x, y, x ∈ R,
3). distributive laws over the addition: x · (y + z) = x · y + x · z and (y + z) · x =
y · x + z · x ∀ x, y, z ∈ R.
We will assume that the ring R has a multiplicative identity element, denoted by 1R , such
that 1R · x = x = x · 1R , ∀ x ∈ R. We also denote the additive identity by 0, i.e., the
identity element of abelian group (R, +).
Definition (Zero divisor). An element x in a ring R is called a zero divisor if there
exists 0 6= y ∈ R such that xy = 0.
Remark.
1). 0 ∈ R is always a zero divisor. We call it trivial zero divisor.
2). There are no nontrivial zero divisors in Z or in polynomial rings Z[t].
Definition (Integral domain). A ring in which 1 6= 0 with no nontrivial zero-divisors
is called an integral domain, i.e.,
xy = 0, x, y ∈ R ⇒ x = 0 or y = 0.
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Math 422.
Coding Theory
Definition (Principal ideal domain). A principal ideal domain (PID) is an integral
domain in which every ideal is principal, i.e., generated by a single element x ∈ R.
Remark.
1). Z. All ideals have the form nZ for some n ∈ Z≥0 .
2). C[t]. But C[t1 , . . . , cn ], n ≥ 2 is not PID.
Definition (Invertible). An element x in a ring R is invertible if ∃ y ∈ R such that
x · y = 1.
Definition (Field). A field is a ring R in which 1 6= 0 and every non-zero element is
invertible.
Theorem. Zm is a field ⇐⇒ m is a prime number.
Definition 2.1 (Finite field and Order of finite field). A finite field is a field F
which has a finite number of elements, this number being called the order of the field,
denoted by |F |.
Theorem 2.1 (Subfield Isomorphic to Zp ). Every finite field has the order of a power
of a prime number p and contains a subfield isomorphic to Zp .
Proof. Let F be a finite field of order n and 1 (one) denote the (unique) multiplicative
identity in F . Consider the ring homomorphism ϕ : Z → F defined by ϕ(n) = n · 1. Z
is a principal ideal domain and F is finite, there is a positive integer p with ker(ϕ) = pZ.
Suppose p = ab with a, b ∈ Z>0 . Then 0 = ϕ(p) = ϕ(ab) = ϕ(a)ϕ(b). Since F is a
field, we have ϕ(a) = 0 or ϕ(b) = 0, i.e., a ∈ ker(ϕ) = pZ or b ∈ ker(ϕ) = pZ. Hence
we have p|a or p|b. On the other hand, we have p = ab. Now we have a = p, b = 1 or
b = p, a = 1, i.e., p is a prime number. The image of ϕ is isomorphic to Z/pZ = Zp ,
a subfield of F . Since ker(ϕ) = pZ, we have p · 1 = ϕ(p) = 0. Now for any element
a ∈ F , we have p · a = p · 1 · a = 0. That is, as an additive abelian group, every nonzero
element in E has order p. If there is a another prime divisor p1 of |F | with p 6= p1 . Then
the Cauchy’s Theorem (or Sylow theorem) gives a nonzero element b ∈ F with p1 b = 0,
contradicting every nonzero element having order p. We conclude that n = |F | = pm for
some m ≥ 1.
Corollary 2.2 (Isomorphism to Zp ). Any field F with prime order p is isomorphic to
Zp .
Proof. The above Theorem says that the prime p must be the power of a prime, which
can only be p itself. It also says that F contains Zp as a subfield. Since the order of Zp
is already p, there are no other elements in F , i.e., F ∼
= Zp .
§.
Theorem 2.2 (Prime Power Fields). There exists a field F of order n.
power of a prime number.
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⇐⇒ n is a
Proof. (⇒) This is implied by above Theorem.
(⇐) Let p be prime and g be an irreducible polynomial of degree r in the polynomial ring
Zp [x] (for a proof of the existence of such a polynomial, see van Lint [1991]). Recall
that every polynomial can be written as a polynomial multiple of g plus a residue
polynomial of degree less than r. The field Zp [x]/hgi, which is just the residue class
polynomial ring Zp [x] (mod g), establishes the existence of a field with exactly pr
elements, corresponding to the p possible choices for each of the r coefficients of a
polynomial of degree less than r.
Remark. From now on, let Fq denote the finite field of order q, where q is a power of
prime number.
Remark (Vector spaces over finite field Fq ). 1). The set Fq n of all ordered n-tuples
over Fq forms a vector space over Fq and its elements will be called vectors. The
addition of vectors and scalar multiplication are given as follows:
a). Addition of vectors: if x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ) ∈ Fqn , then
x + y = (x1 + y1 , x2 + y2 , . . . , xn + yn ).
b). Scalar multiplication: if x = (x1 , x2 , . . . , xn ) ∈ Fqn and a ∈ Fq , then ax =
(ax1 , ax2 , . . . , axn ).
2). dim Fqn = n and Fqn contains exactly q n vectors.
3). We can also define the subspace C of vector space Fqn . It is clear that a subset C
of Fqn is a subspace of it if and only if C is closed under linear combinations. If
dim(C) = k, then C contains q k vectors.