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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. 25 26 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. 27 ⇐⇒ 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.