Introduction to Finite Fields

Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in finite fields, i.e., fields with a finite number of elements (also called Galois fields). In the next chapter, finite fields will be used to develop Reed-Solomon (RS) codes, the most useful class of algebraic codes. Groups and polynomials provide the requisite background to understand finite fields. A field is more than just a set of elements: it is a set of elements under two operations, called addition and multiplication, along with a set of properties governing these operations. The addition and multiplication operations also imply inverse operations called subtraction and division. The reader is presumably familiar with several examples of fields, such as the real field R, the complex field C, the field of rational numbers Q, and the binary field F2 . 7.1 Summary In this section we briefly summarize the results of this chapter. The main body of the chapter will be devoted to defining and explaining these concepts, and to proofs of these results. For each prime p and positive integer m ≥ 1, there exists a finite field Fpm with pm elements, and there exists no finite field with q elements if q is not a prime power. Any two fields with pm elements are isomorphic. The integers modulo p form a prime field Fp under mod-p addition and multiplication. The polynomials Fp [x] over Fp modulo an irreducible polynomial g(x) ∈ Fp [x] of degree m form a finite field with pm elements under mod-g(x) addition and multiplication. For every prime p, there exists at least one irreducible polynomial g(x) ∈ Fp [x] of each positive degree m ≥ 1, so all finite fields may be constructed in this way. Under addition, Fpm is isomorphic to the vector space (Fp )m . Under multiplication, the nonzero m elements of Fpm form a cyclic group {1, α, . . . , αp −2 } generated by a primitive element α ∈ Fpm . m The elements of Fpm are the pm roots of the polynomial xp − x ∈ Fp [x]. The polynomial − x is the product of all monic irreducible polynomials g(x) ∈ Fp [x] such that deg g(x) divides m. The roots of a monic irreducible polynomial g(x) ∈ Fp [x] form a cyclotomic coset of deg g(x) elements of Fpm which is closed under the operation of raising to the pth power. m xp For every n that divides m, Fpm contains a subfield with pn elements. For further reading on this beautiful subject, see [E. R. Berlekamp, Algebraic Coding Theory, Aegean Press, 1984], [R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge University Press, 1986] or [R. J. McEliece, Finite Fields for Computer Scientists and Engineers, Kluwer, 1987], [M. R. Schroeder, Number Theory in Science and Communication, Springer, 1986], or indeed any book on finite fields or algebraic coding theory. 7.2 The integers We begin with a brief review of the familiar factorization properties of the set Z of integers. We will use these properties immediately in our discussion of cyclic groups and their subgroups and of prime fields. Moreover, we will model our later discussion of the factorization properties of polynomials on the discussion here. 7.2.1 Definitions An integer n is said to be a divisor of an integer i if i is an integer multiple of n; i.e., i = qn for some integer q. Thus all integers are trivially divisors of 0. The integers that have integer inverses, namely ±1, are called the units of Z. If u is a unit and n is a divisor of i, then un is a divisor of i and n is a divisor of ui. Thus the factorization of an integer can only be unique up to a unit u, and ui has the same divisors as i. We therefore consider only factorizations of positive integers into products of positive integers. Every nonzero integer i is divisible by 1 and i; these divisors are called trivial. An integer n is said to be a factor of an integer i if n is positive and a nontrivial divisor of i. For example, 1 has no nontrivial divisors and thus no factors. A positive integer that has no nontrivial divisors is called a prime integer. 7.2.2 Mod-n arithmetic Given a positive integer n, every integer i may be uniquely expressed as i = qn + r for some integer remainder r in the interval 0 ≤ r ≤ n − 1 and some integer quotient q. This may be proved by the Euclidean division algorithm, which if i ≥ n just subtracts n from i repeatedly until the remainder lies in the desired interval. The remainder r, denoted by r = i mod n, is the more important part of this expression. The set of possible mod-n remainders is the set of n integers Rn = {0, 1, . . . , n − 1}. Evidently n is a divisor of i if and only if i mod n = 0. Remainder arithmetic using the mod-n remainder set Rn is called “mod-n arithmetic.” The rules for mod-n arithmetic follow from the rules for integer arithmetic as follows. Let r = i mod n and s = j mod n; then, as integers, r = i − qn and s = j − tn for some quotients q and t. Then r + s = i + j − (q + t)n; rs = ij − (qj + ti)n + qtn2 . Hence (r + s) mod n = (i + j) mod n and rs mod n = ij mod n; i.e., the mod-n remainder of the sum or product of two integers is equal to the mod-n remainder of the sum or product of their mod-n remainders, as integers. The mod-n addition and multiplication rules are therefore defined as follows: r ⊕ s = (r + s) mod n; r ∗ s = (rs) mod n, where “r” and “s” denote elements of the remainder set Rn on the left and the corresponding ordinary integers on the right. This makes mod-n arithmetic consistent with ordinary integer arithmetic in the sense expressed in the previous paragraph. 7.2.3 Unique factorization Given a positive integer i, we may factor i into a unique product of prime factors by simply factoring out primes no greater than i until we arrive at the quotient 1, as the reader has known since grade school. For the time being, we will take this unique factorization property as given. A proof will be given as an exercise after we prove the corresponding property for polynomials. 7.3 Groups We now introduce groups. Definition 7.1 A group is a set of elements G = {a, b, c, . . .} and an operation ⊕ for which the following axioms hold: • Closure: for any a ∈ G, b ∈ G, the element a ⊕ b is in G. • Associative law: for any a, b, c ∈ G, (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c). • Identity: There is an identity element 0 in G for which a ⊕ 0 = 0 ⊕ a = a for all a ∈ G. • Inverse: For each a ∈ G, there is an inverse (−a) such that a ⊕ (−a) = 0. In general it is not necessary that a ⊕ b = b ⊕ a. A group G for which a ⊕ b = b ⊕ a for all a, b ∈ G is called abelian or commutative. In these notes all groups will be abelian. In view of the associative law, we may write (a⊕b)⊕c as a⊕b⊕c without ambiguity. Moreover, in an abelian group the elements a, b, c may be written in any order. Frequently, the operation in a group is called multiplication, usually represented either by ∗ or juxtaposition. The identity is then denoted by 1 (or e) and the inverse of a by a−1 . Additive notation is generally used only for abelian groups, whereas multiplicative notation is used for both abelian and nonabelian groups. Since we consider only abelian groups, we will use additive notation when the nature of the group is unspecified. As an example, the set of integers Z with the usual addition operation + forms an abelian group. Also, the real field R forms an additive abelian group under ordinary addition in which the identity is 0 and the inverse of a is −a. More interestingly, as the reader should verify, the nonzero elements of R form a multiplicative abelian group under ordinary multiplication, in which the identity is 1 and the inverse of a is a−1 = 1/a. We will see that every field has similar additive and multiplicative group properties. This example illustrates that the group structure (i.e., the properties stemming from the group operation ⊕) may reflect only part of the structure of the given set of elements; e.g., the additive group structure of R takes no account of the fact that real numbers may also be multiplied, and the multiplicative group structure of R − {0} takes no account of the fact that real numbers may also be added. We abbreviate b ⊕ (−a) for any a, b ∈ G by b − a and regard “−” as an additional operation implicitly defined by the axioms. In an additive group, “−” is called subtraction; in a multiplicative group, “−” is called division and denoted by / or ÷. Because of the inverse operation, cancellation is always permissible; i.e., if x ⊕ a = y ⊕ a, we can add −a to both sides, showing that x = y. Similarly, one can move terms from one side of an equation to the other; i.e., x ⊕ a = y implies x = y − a. Exercise 1 (Inverses and cancellation) (a) Verify the following set of implications for arbitrary elements a, b of a group G which is not necessarily abelian: b ⊕ a = 0 ⇒ b = −a ⇒ a ⊕ b = 0 ⇒ a = −b ⇒ b ⊕ a = 0. (b) Use this result to show that the inverse is unique, i.e., that a ⊕ b = 0 ⇒ b = −a, and that the inverse also works on the left, i.e., b ⊕ a = 0 ⇒ b = −a. Note that this shows that cancellation is permitted on either the right or the left. (c) Show that the identity element is unique, i.e., that for a, b ∈ G, a ⊕ b = a ⇒ b = 0 and b ⊕ a = a ⇒ b = 0. If G has a finite number of elements, G = {a1 , a2 , . . . , an }, then G is said to be finite and |G| = n is said to be the order of G. The group operation ⊕ may then be specified by an n × n “addition table” whose entry at row i, column j is ai ⊕ aj . The cancellation property implies that if aj = ak , then ai ⊕ aj =  ai ⊕ ak . This means that all elements in any row i of the addition table are distinct; i.e., each row contains each element of G exactly once. Similarly, each column contains each element of G exactly once. Thus the group axioms restrict the group operation ⊕ more than might be immediately evident. 7.3.1 Alternative group axioms The property that a “row of the addition table,” namely a ⊕ G = {a ⊕ b | b ∈ G} is just the set of elements of G in a different order (i.e., a permutation of G) is a fundamental property of any group G. We will now show that this permutation property may be taken as one of the group axioms. Subsequently we will use this property to prove that certain sets are groups. Theorem 7.1 (Alternative group axioms) Let G = {a, b, c, . . .} be a set of elements on which an operation ⊕ is defined. Then G is a group under the operation ⊕ if and only if the following axioms hold: • Associative law: for any a, b, c ∈ G, (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c). • Identity: There is an identity element 0 in G for which a ⊕ 0 = 0 ⊕ a = a for all a ∈ G. • Permutation property: For each a ∈ G, a ⊕ G = {a ⊕ b | b ∈ G} is a permutation of G. Proof. (⇒) If G is a group under ⊕, then by the closure property every element a ⊕ b is in G. Moreover, the fact that a ∈ G has an inverse −a ∈ G implies that every element b ∈ G may be written as a ⊕ (−a ⊕ b) ∈ a ⊕ G, so every element of G is in a ⊕ G. Finally, from the cancellation property, a ⊕ b = a ⊕ c implies b = c. Thus the correspondence between G and a ⊕ G defined by b ↔ a ⊕ b is one-to-one; i.e., a permutation. (⇐) Conversely, if a ⊕ G is a permutation of G for every a ∈ G, then (a) the closure property holds; i.e., a ⊕ b ∈ G for all a, b ∈ G; (b) since 0 ∈ a ⊕ G, there must exist a unique b ∈ G such that a ⊕ b = 0, so a has a unique inverse −a = b under ⊕. Thus G is a group under ⊕. The properties of “rows” a ⊕ G hold equally for “columns” G ⊕ a, even when G is nonabelian. For example, the set R∗ of nonzero elements of the real field R form an abelian group under real multiplication, because real multiplication is associative and commutative with identity 1, and αR∗ is a permutation of R∗ for any α ∈ R∗ . Exercise 2 (Invertible subsets). (a) Let H be a set of elements on which an associative operation ⊕ is defined with identity 0, and let G be the subset of elements h ∈ H which have unique inverses −h such that h ⊕ −h = 0. Show that G is a group under ⊕. (b) Show that the nonzero elements of the complex field form a group under complex multiplication. (c) Show that the set of invertible n × n real matrices forms a (nonabelian) group under real matrix multiplication. (d) What are the invertible elements of Z under multiplication? Do they form a group? 7.3.2 Cyclic groups An important example of a finite abelian group is the set of remainders Rn = {0, 1, . . . , n − 1} under mod-n addition, where n is a given positive integer. This group is called “the integers mod n” and is denoted by Zn . Note that Z1 is the trivial group {0}. A finite cyclic group is a finite group G with a particular element g ∈ G, called the generator, such that each element of G can be expressed as the sum, g⊕· · ·⊕g, of some number of repetitions of g.1 Thus each element of G appears in the sequence of elements {g, g ⊕ g, g ⊕ g ⊕ g, . . .}. We denote such an i-fold sum by ig, where i is a positive integer and g is a group element; i.e., 1g = g, 2g = g ⊕ g, . . . , ig = g ⊕ · · · ⊕ g, . . . � �� � i terms Since g generates G and G includes the identity element 0, we must have ig = 0 for some positive integer i. Let n be the smallest such integer; thus ng = 0 and ig = 0 for 1 ≤ i ≤ n − 1. Adding the sum of j g’s for any j > 0 to each side of ig = 0 results in (i + j)g =  jg. Thus the elements {1g, 2g, . . . , ng = 0} must all be different. 1 Mathematicians say also that an infinite group G = {. . . , −1g, 0g, 1g, 2g, . . .} generated by a single element g is cyclic; e.g., the group of integers Z is an infinite cyclic group with generator 1. Although such infinite cyclic groups have the single-generator property of finite cyclic groups, they do not “cycle.” Hereafter, “cyclic group” will mean “finite cyclic group.” We can also add jg to both sides of the equality ng = 0, yielding (j + n)g = jg for any j > 0. Thus for each i > n, ig is equal to some earlier element in the sequence, namely (i − n)g. The elements {1g, 2g, . . . , ng = 0} therefore constitute all of the distinct elements in G, and the order of G is |G| = n. If we define 0g to be the identity 0, then the elements of G may be conveniently represented as G = {0g = 0, 1g, . . . , (n − 1)g}. Figure 1 illustrates the cyclic structure of G that arises from the relation (j + n)g = jg. 0 = ng = 2ng = · · · r r (n − 1)g = r g = (n + 1)g = · · · (2n − 1)g = · · · r 2g = (n + 2)g = · · · r r r r 3g = (n + 3)g = · · · 4g = (n + 4)g = · · · Figure 1. The cyclic structure of a cyclic group: the sequence {1g, 2g, . . .} goes from the group element g up to ng = 0, then returns to g and continues to cycle. Addition in a cyclic group of order n can be understood in terms of mod-n addition. In particular, since ng = 0, we also have 2ng = 0, 3ng = 0, etc. Since any integer i may be uniquely written as i = qn + r where the remainder r = i mod n is in the set Rn = {0, 1, . . . , n − 1}, we have ig = (qn)g + rg = rg, where rg = (i mod n)g is one of the elements of G. The addition rule of G is thus as follows: for each 0 ≤ i, j < n, ig ⊕ jg = (i + j mod n)g. Evidently 0g is the identity, and the inverse of a nonzero element ig is (n − i)g. We thus see that any cyclic group G of order n is essentially identical to Zn . More precisely, the correspondence ig ∈ G ↔ i ∈ Zn is preserved under addition; i.e., ig ⊕ jg ↔ i ⊕ j for each i, j ∈ Zn . This type of correspondence is called an isomorphism. Specifically, two finite groups G and H are isomorphic if there exists an invertible2 function h : G → H mapping each α ∈ G into a β = h(α) ∈ H such that h(α ⊕ α ) = h(α) ⊕ h(α ), where ⊕ denotes the group operation of G on the left and that of H on the right. In summary: Theorem 7.2 (Cyclic groups) The elements of a cyclic group G of order n with generator g are {0g, 1g, 2g, . . . , (n − 1)g}. The addition rule is ig ⊕ jg = (i + j mod n)g, the identity is 0g, and the inverse of ig = 0g is (n − i)g. Finally, G is isomorphic to Zn under the one-to-one correspondence ig ↔ i. Since Zn is abelian, it follows that all cyclic groups are abelian. In multiplicative notation, the elements of a cyclic group G of order n with generator g are denoted by {g 0 = 1, g 1 , g 2 , . . . , g n−1 }, the multiplication rule is g i ∗g j = g (i+j mod n) , the identity is g 0 = 1, and the inverse of g i =  1 is g n−i . For example, if ω = e2πi/n , the set {1, ω, ω 2 , . . . , ω n−1 } of complex nth roots of unity is a cyclic group under complex multiplication, isomorphic to Zn . 2 A function h : G → H is called invertible if for each β ∈ H there is a unique α ∈ G such that β = h(α). An invertible function is also called a one-to-one correspondence, denoted by G ↔ H. 7.3.3 Subgroups A subgroup S of a group G is a subset of the elements of the group such that if a, b ∈ S, then a ⊕ b ∈ S and −a ∈ S. A subgroup S therefore includes the identity element of G and the inverse of each element in S. The associative law holds for S since it holds for G. Therefore a subgroup S ⊆ G is itself a group under the group operation of G. For example, the set of integers Z is a subgroup of the additive group of R. If G is abelian, then S must be abelian; however, S may be abelian even if G is nonabelian. For any g ∈ G, we define the coset (translate) S ⊕ g = {s ⊕ g | s ∈ S}. The zero coset S ⊕ 0 is thus equal to S itself; moreover, by Theorem 7.1, S ⊕ g = S whenever g ∈ S. The following theorem states a more general result: Lemma 7.3 (Cosets) Two cosets S ⊕ g and S ⊕ h are the same if g − h ∈ S, but are disjoint if g − h ∈ / S. Proof. If g − h ∈ S, then the elements of S ⊕ h include (g − h) ⊕ h = g and therefore all elements of S ⊕ g, so S ⊕ g ⊆ S ⊕ h; similarly S ⊕ h ⊆ S ⊕ g. On the other hand, if S ⊕ g and S ⊕ h have any element in common, say s ⊕ g = s ⊕ h, then g − h = s − s ∈ S; thus, g − h ∈ / S implies that S ⊕ g and S ⊕ h are disjoint. It follows that the distinct cosets S ⊕ g of a subgroup S ⊆ G form a disjoint partition of G, since every element g ∈ G lies in some coset, namely S ⊕ g. The elements s ⊕ g of a coset S ⊕ g are all distinct, since s ⊕ g = s ⊕ g implies s = s . Therefore if S is finite, then all cosets of S have the same size, namely the size |S| of S = S ⊕ 0. If G is finite, G is therefore the disjoint union of a finite number |C| of cosets of S ⊆ G, each of size |S|, so |G| = |C||S|. This proves Lagrange’s theorem: Theorem 7.4 (Lagrange) If S is a subgroup of a finite group G, then |S| divides |G|. 7.3.4 Cyclic subgroups Given any finite group G and any element g ∈ G, the set of elements generated by g, namely S(g) = {g, g ⊕ g, . . .}, is a cyclic subgroup of G. The order of g is defined as the order |S(g)| of S(g). By Lagrange’s theorem, |S(g)| divides |G|, and by the cyclic groups theorem, S(g) is isomorphic to Z|S(g)| . (If g = 0, then S(g) = {0} and |S(g)| = 1. We will assume g = 0.) As a fundamental example, let G be the cyclic group Zn = {0, 1, . . . , n − 1}, and let S(m) be the cyclic subgroup {m, 2m, . . .} generated by m ∈ Zn . Here im = m ⊕ · · · ⊕ m is simply the sum of m with itself i times; i.e., im ∈ G is the ordinary product im mod n. The order |S(m)| of S(m) is the least positive integer k such that km = 0 mod n; i.e., such that the integer product km is divisible by n. Thus km is the least common multiple of m and n, denoted lcm(m, n), and |S(m)| = k = lcm(m, n)/m. By elementary number theory, lcm(m, n) = mn/ gcd(m, n) for any positive integers m, n, so we may alternatively write |S(m)| = n/ gcd(m, n), where gcd(m, n) denotes the greatest common divisor of m and n. This shows explicitly that |S(m)| divides n. For example, suppose n = 10 and m = 4. Then S(4) = {4, 8, 2, 6, 0}. Thus |S(4)| = 5, consistent with |S(4)| = lcm(4, 10)/4 = 20/4 or |S(4)| = 10/ gcd(4, 10)/4 = 10/2. Now when does S(m) = Zn ? This occurs if and only if gcd(m, n) = 1; i.e., if and only if m is relatively prime to n. In short, m generates Zn and has order |S(m)| = n if and only if m and n are relatively prime. The number of integers in the set {0, 1, . . . , n − 1} that have order n is called the Euler number φ(n). For example, in Z10 the integers that are relatively prime to 10 are {1, 3, 7, 9}, so φ(10) = 4. The order of the other elements of Z10 are as follows: • 0 is the only element of order 1, and S(0) = {0}. • 5 is the only element of order 2, and S(5) = {0, 5}. • {2, 4, 6, 8} have order 5, and S(2) = S(4) = S(6) = S(8) = {0, 2, 4, 6, 8}. In general, Zn has a cyclic subgroup Sd of order d for each positive integer d that divides n, including 1 and n. Sd consists of {0, n/d, 2n/d, . . . , (d −1)n/d}, and is isomorphic to Zd . Sd thus contains φ(d) elements that are relatively prime to d, each of which has order d and generates Sd . The remaining elements of Sd belong also to smaller cyclic subgroups. For example, Z10 has a subgroup S5 = {0, 2, 4, 6, 8} with 5 elements. Four of these elements, namely {2, 4, 6, 8}, are relatively prime to 5 and generate S5 . The remaining element of S5 , namely 0, has order 1. Since every element of Zn has some definite order d that divides n, we have � n= φ(d). (7.1) d: d|n The notation d : d|n means the set of positive integers d, including 1 and n, that divide n. All Euler numbers may be determined recursively from this expression. For example, φ(1) = 1, φ(2) = 2 − φ(1) = 1, φ(3) = 3 − φ(1) = 2, φ(4) = 4 − φ(1) − φ(2) = 2, . . .. Exercise 3. Show that φ(n) ≥ 1 for all n ≥ 1. [Hint: Find the order of 1 in Zn .] Since every cyclic group G of size n is isomorphic to Zn , these results apply to every cyclic group. In particular, every cyclic group G of size n has φ(n) generators that generate G, which are called the primitive elements of G. G also contains one cyclic subgroup of size d for each d that divides n. Exercise 4. Show that every subgroup of Zn is cyclic. [Hint: Let s be the smallest nonzero element in a subgroup S ⊆ Zn , and compare S to the subgroup generated by s.] 7.4 Fields Definition 7.2 A field is a set F of at least two elements, with two operations ⊕ and ∗, for which the following axioms are satisfied: • The set F forms an abelian group (whose identity is called 0) under the operation ⊕. • The set F∗ = F − {0} = {a ∈ F, a = 0} forms an abelian group (whose identity is called 1) under the operation ∗. • Distributive law: For all a, b, c ∈ F, (a ⊕ b) ∗ c = (a ∗ c) ⊕ (b ∗ c). The operation ⊕ is called addition (and often denoted by +), and the operation ∗ is called multiplication (and often denoted by juxtaposition). As in ordinary arithmetic, we often omit the parentheses around a product of elements, using the convention “multiplication before addition;” e.g., we interpret a ⊕ b ∗ c as a ⊕ (b ∗ c). The reader may verify that R, C, Q and F2 each form a field according to this definition under conventional addition and multiplication. Exercise 5. Show that for any element a ∈ F, a ∗ 0 = 0. 7.4.1 Prime fields A fundamental example of a finite (Galois) field is the set Fp of mod-p remainders, where p is a given prime number. Here, as in Zp , the set of elements is Rp = {0, 1, · · · , p − 1}, and the operation ⊕ is mod-p addition. The multiplicative operation ∗ is mod-p multiplication; i.e., multiply integers as usual and then take the remainder after division by p. Theorem 7.5 (Prime fields) For every prime p, the set Rp = {0, 1, · · · , p − 1} forms a field (denoted by Fp ) under mod-p addition and multiplication. Proof. We have already seen that the elements of Fp form an abelian group under addition modulo p, namely the cyclic group Zp . The associative and commutative properties of multiplication mod p follow from the corresponding properties of ordinary multiplication; the distributive law follows from the corresponding property for ordinary addition and multiplication. The multiplicative identity is 1. To see that the nonzero elements F∗p = Fp − {0} form a group under multiplication, we use Theorem 7.1. By unique factorization, the product of two nonzero integers a, b < p cannot equal 0 mod p. Therefore the nonzero elements F∗p are closed under multiplication mod p. Also, for a, b, c ∈ F∗p and b =  c we have a(b − c) mod p = 0. Thus ab =  ac mod p, which implies a ∗ b = a ∗ c. Consequently there are no zeroes or repetitions in the set of p − 1 elements {a ∗ 1, a ∗ 2, . . . , a ∗ (p − 1)}, which means they must be a permutation of F∗p . We next show that Fp is essentially the only field with p elements. More precisely, we show that all fields with p elements are isomorphic. Two fields F and G are isomorphic if there is an invertible function h : F → G mapping each α ∈ F into a β = h(α) ∈ G such that h(α ⊕ α ) = h(α) ⊕ h(α ) and h(α ∗ α ) = h(α) ∗ h(α ). Less formally, F and G are isomorphic if there is a one-to-one correspondence F ↔ G that translates the addition and multiplication tables of F to those of G and vice versa. Let F be any field with a prime p number of elements. By the field axioms, F has an additive identity 0 and multiplicative identity 1. Consider the additive cyclic subgroup generated by 1, namely S(1) = {1, 1 ⊕ 1, . . .}. By Lagrange’s theorem, the order of S(1) divides |F| = p, and therefore must be equal to 1 or p. But 1 ⊕ 1 = 1, else 1 = 0, so 1 must have order p. In other words, S(1) = F, and the additive group of F is isomorphic to that of Zp . We may therefore denote the elements of F by {0, 1, 2, . . . , p − 1}, and use mod-p addition as the addition rule. The only remaining question is whether this correspondence F ↔ Zp under addition extends to multiplication. The distributive law shows that it does: j ∗ i is the sum of j terms each equal to i, so j ∗ i = (ji mod p). Therefore, in summary: Theorem 7.6 (Prime field uniqueness) Every field F with a prime number p of elements is isomorphic to Fp via the correspondence 1� ⊕ ·�� · · ⊕ 1� ∈ F ↔ i ∈ Fp . i terms In view of this elementary isomorphism, we will denote any field with a prime number p of elements by Fp . It is important to note that the set Zn of integers mod n does not form a field if n is not prime. The reason is that n = ab for some positive integers a, b < n ∈ Zn ; thus ab = 0 mod n, so the set of nonzero elements of Zn is not closed under multiplication mod n. However, we will see shortly that there do exist finite fields with non-prime numbers of elements that use other rules for addition and multiplication. 7.4.2 The prime subfield of a finite field A subfield G of a field F is a subset of the field that is itself a field under the operations of F. For example, the real field R is a subfield of the complex field C. We now show that every finite field Fq has a subfield that is isomorphic to a prime field Fp . Let Fq be a finite field with q elements. By the field axioms, Fq has an additive identity 0 and a multiplicative identity 1. Consider the cyclic subgroup of the additive group of Fq that is generated by 1, namely S(1) = {1, 1 ⊕ 1, . . .}. Let n = |S(1)|. By the cyclic group theorem, S(1) is isomorphic to Zn , and its elements may be denoted by {0, 1, 2, . . . , n − 1}, with mod-n addition. By the distributive law in Fq , the product i∗j (in Fq ) of two nonzero elements in S(1) is simply the sum of ij ones, which is an element of S(1), namely ij mod n. Since this is a product of nonzero elements of Fq , by the field axioms ij mod n must be nonzero for all nonzero i, j. This will be true if and only if n is a prime number p. Thus S(1) forms a subfield of Fq with a prime number p of elements. By the prime field theorem of the previous subsection, S(1) is isomorphic to Fp . Thus the elements of S(1), which are called the integers of Fq , may be denoted by Fp = {0, 1, . . . , p − 1}, and the addition and multiplication rules of Fq reduce to mod-p addition and multiplication in Fp . The prime p is called the characteristic of Fq . Since the p-fold sum of the identity 1 with itself is 0, the p-fold sum of every field element β ∈ Fq with itself is 0: pβ = 0. In summary: Theorem 7.7 (Prime subfields) The integers {1, 1 ⊕ 1, . . .} of any finite field Fq form a subfield Fp ⊆ Fq with a prime number p of elements, where p is the characteristic of Fq . 7.5 Polynomials We now consider polynomials over Fp , namely polynomials whose coefficients lie in Fp and for which polynomial addition and multiplication is performed in Fp . We will see that the factorization properties of polynomials are similar to those of the integers, and that the analogue to mod-n arithmetic is arithmetic modulo a polynomial f (x). A nonzero polynomial f (x) of degree m over a field F is an expression of the form f (x) = f0 + f1 x + f2 x2 + · · · + fm xm , where fi ∈ F, 0 ≤ i ≤ m, and fm = 0. We say that deg f (x) = m. The symbol x represents an indeterminate (or “placeholder”), not an element of F; i.e., two polynomials are different if and only if their coefficients are different3 . The nonzero polynomials of degree 0 are simply the nonzero field elements f0 ∈ F. There is also a special zero polynomial f (x) = 0 whose degree is defined by convention as deg 0 = −∞; we will explain the reason for this convention shortly. The set of all polynomials over F in an indeterminate x is denoted by F[x]. The rules for adding, subtracting or multiplying polynomials are the same over a general field F as over the real field R, except that coefficient operations are in F. In particular, addition and subtraction are performed componentwise. For multiplication, the coefficients of a polynomial product f (x) = h(x)g(x) are determined by convolution: fi = i � hj gi−j . j=0 If two nonzero polynomials are multiplied, then their degrees add; i.e., deg(h(x)g(x)) = deg h(x) + deg g(x). The convention deg 0 = −∞ ensures that this formula continues to hold when h(x) or g(x) is the zero polynomial. The set F[x] has many of the properties of a field. It is evidently an abelian group under addition whose identity is the zero polynomial 0 ∈ F[x]. It is closed under multiplication, which is both associative and commutative and which distributes over addition. It has a multiplicative identity 1 ∈ F[x], and the cancellation law holds. However, in general we cannot divide evenly by a nonzero polynomial, since a polynomial f (x) with deg f (x) > 0 has no multiplicative inverse. Therefore F[x] is a ring,4 not a field, like the ring of integers Z. We now develop a series of properties of F[x] that resemble those of Z. 3 Over the real field R, a polynomial f (x) is sometimes regarded as a function f : R → R. This alternative viewpoint makes little difference in the real case, since two polynomials over R are different if and only if the corresponding polynomial functions are different. However, over finite fields it is important to maintain the distinction. For example, over F2 the polynomial functions x and x2 both map 0 → 0, 1 → 1, yet the polynomials x and x2 are different. 4 The axioms of a ring are similar to those for a field, except that there is no multiplicative inverse. For example, Z and Zn (for n not a prime) are rings. In fact, Z and F[x] are integer domains, which are the nicest kind of rings. An integer domain is a ring with commutative multiplication and a multiplicative identity 1 such that the nonzero elements are closed under multiplication. Exercise 6. Show that an integer domain with a finite number of elements must be a finite field. [Hint: consider its cyclic multiplicative subgroups.] 7.5.1 Definitions A polynomial g(x) is said to be a divisor of an polynomial f (x) if f (x) is a polynomial multiple of g(x); i.e., f (x) = q(x)g(x) for some polynomial q(x). Thus all polynomials are trivially divisors of the zero polynomial 0. The polynomials that have polynomial inverses are the nonzero degree-0 polynomials β ∈ F∗ = F − {0}. These are called the units of F[x]. If u(x) is a unit polynomial and g(x) is a divisor of f (x), then u(x)g(x) is a divisor of f (x) and g(x) is a divisor of u(x)f (x). Thus the factorization of a polynomial can be unique only up to a unit polynomial u(x), and u(x)f (x) has the same divisors as f (x). A monic polynomial is a nonzero polynomial f (x) of degree m with high-order coefficient fm equal to 1; i.e., f (x) = f0 + f1 x + f2 x2 + · · · + xm . Every nonzero polynomial g(x) may be written as the product g(x) = gm f (x) of a monic polynomial f (x) of the same degree with a unit polynomial u(x) = gm , and the product of two monic polynomials is monic. We may therefore consider only factorizations of monic polynomials into products of monic polynomials. Every nonzero polynomial f (x) is divisible by 1 and f (x); these divisors are called trivial. A polynomial g(x) is said to be a factor of a polynomial f (x) if g(x) is monic and a nontrivial divisor of f (x). Thus the degree of any factor g(x) of f (x) satisfies 1 ≤ deg g(x) < deg f (x). A polynomial g(x) of degree 1 or more that has no factors is called an irreducible polynomial, and a monic irreducible polynomial is called a prime polynomial. Our goal now is to show that every monic polynomial has a unique factorization into prime polynomial factors. 7.5.2 Mod-g(x) arithmetic Given a monic polynomial g(x) of degree m, every polynomial f (x) may be expressed as f (x) = q(x)g(x)+r(x) for some polynomial remainder r(x) such that deg r(x) < m and some polynomial quotient q(x). This may be proved by the Euclidean long division algorithm of high school, with component operations in F; i.e., divide g(x) into f (x) by long division, high-degree terms first, stopping when the degree of the remainder is less than that of g(x). The following exercise shows that the resulting quotient q(x) and remainder r(x) are unique. Exercise 7 (Euclidean division algorithm). (a) For the set F[x] of polynomials over any field F, show that the distributive law holds: (f1 (x) + f2 (x))h(x) = f1 (x)h(x) + f2 (x)h(x). (b) Use the distributive law to show that for any given f (x) and g(x) in F[x], there is a unique q(x) and r(x) with deg r(x) < deg g(x) such that f (x) = q(x)g(x) + r(x). The remainder polynomial r(x), denoted by r(x) = f (x) mod g(x), is the more important part of this decomposition. The set of all possible remainder polynomials is the set RF,m = {r0 + r1 x + · · · + rm−1 xm−1 | rj ∈ F, 0 ≤ j ≤ m − 1}, whose size is |RF,m | = |F|m . Evidently g(x) is a divisor of f (x) if and only if f (x) mod g(x) = 0. Remainder arithmetic using the remainder set RF,m is called “mod-g(x) arithmetic.” The rules for mod-g(x) arithmetic follow from the rules for polynomial arithmetic as follows. Let r(x) = f (x) mod g(x) and s(x) = h(x) mod g(x); then, as polynomials, r(x) = f (x) − q(x)g(x) and s(x) = h(x) − t(x)g(x) for some quotient polynomials q(x) and t(x). Then f (x) + h(x) = r(x) + s(x) − (q(x) + t(x))g(x); f (x)h(x) = r(x)s(x) − (q(x)s(x) + t(x)r(x))g(x) + q(x)t(x)g 2 (x). Hence (f (x) + h(x)) mod g(x) = (r(x) + s(x)) mod g(x) and f (x)h(x) mod g(x) = r(x)s(x) mod g(x). In other words, the mod-g(x) remainder of the sum or product of two polynomials is equal to the mod-g(x) remainder of the sum or product of their mod-g(x) remainders. The mod-g(x) addition and multiplication rules are therefore defined as follows: r(x) ⊕ s(x) = (r(x) + s(x)) mod g(x); r(x) ∗ s(x) = (r(x)s(x)) mod g(x), where “r(x)” and “s(x)” denote elements of the remainder set RF,m on the left and the corresponding ordinary polynomials on the right. This makes mod-g(x) arithmetic consistent with ordinary polynomial arithmetic in the sense of the previous paragraph. Note that the mod-g(x) addition rule is just componentwise addition of coefficients in F. In this sense the additive groups of RF,m and of the vector space Fm of m-tuples over F are isomorphic. 7.5.3 Unique factorization By definition, every monic polynomial f (x) is either irreducible or can be factored into a product of monic polynomial factors, each of lower degree. In turn, if a factor is not irreducible, it can be factored further. Since factor degrees are decreasing but bounded below by 1, we must eventually arrive at a product of monic irreducible (prime) polynomials. The following theorem shows that there is only one such set of prime polynomial factors, regardless of the order in which the polynomial is factored. Theorem 7.8 (Unique factorization of polynomials) Over any field F, every monic polynomial f (x) ∈ F[x] of degree m ≥ 1 may be written in the form f (x) = k � ai (x), i=1 where each ai (x), 1 ≤ i ≤ k, is a prime polynomial in F[x]. This factorization is unique, up to the order of the factors. Proof. We have already shown that f (x) may be factored in this way, so we need only prove uniqueness. Thus assume hypothetically that the theorem is false and let m be the smallest degree such that there exists a degree-m monic polynomial f (x) with more than one such factorization, f (x) = a1 (x) · · · ak (x) = b1 (x) · · · bj (x); j, k ≥ 1, (7.2) where a1 (x), . . . , ak (x) and b1 (x), . . . , bj (x) are prime polynomials. We will show that this implies a polynomial f  (x) with degree less than m with non-unique factorization, and this contradiction will prove the theorem. Now a1 (x) cannot appear on the right side of (7.2), else it could be factored out for an immediate contradiction. Similarly, b1 (x) cannot appear on the left. Without loss of generality, assume deg b1 (x) ≤ deg a1 (x). By the Euclidean division algorithm, a1 (x) = q(x)b1 (x) + r(x). Since a1 (x) is irreducible, r(x) = 0 and 0 ≤ deg r(x) < deg b1 (x) ≤ deg a1 (x). Thus r(x) has a prime factorization r(x) = βr1 (x) · · · rn (x), where β is the high-order coefficient of r(x), and b1 (x) is not a divisor of any of the ri (x), since it has greater degree. Substituting into (7.2), we have (q(x)b1 (x) + βr1 (x) · · · rn (x))a2 (x) · · · ak (x) = b1 (x) · · · bj (x), or, defining f  (x) = r1 (x) · · · rn (x)a2 (x) · · · ak (x) and rearranging terms, f  (x) = r1 (x) · · · rn (x)a2 (x) · · · ak (x) = β −1 b1 (x)(b2 (x) · · · bj (x) − q(x)a2 (x) · · · ak (x)). Now f  (x) is monic, because it is a product of monic polynomials; it has degree less than f (x), since deg r(x) < deg a1 (x); and it has two different factorizations, with b1 (x) a factor in one but not a divisor of any of the factors in the other; contradiction. Exercise 8. Following this proof, prove unique factorization for the integers Z. 7.5.4 Enumerating prime polynomials The prime polynomials in F[x] are analogous to the prime numbers in Z. One way to enumerate the prime polynomials is to use an analogue of the sieve of Eratosthenes. For integers, this method goes as follows: Start with a list of all integers greater than 1. The first integer on the list is 2, which is prime. Erase all multiples of 2 (even integers). The next remaining integer is 3, which must be the next prime. Erase all multiples of 3. The next remaining integer is 5, which must be the next prime. Erase all multiples of 5. And so forth. Similarly, to find the prime polynomials in F2 [x], for example, first list all polynomials of degree 1 or more in F2 [x] in order of degree. (Note that all nonzero polynomials in F2 [x] are monic.) No degree-1 polynomial can have a factor, so the two degree-1 polynomials, x and x + 1, are both prime. Next, erase all degree-2 multiples of x and x + 1, namely x2 = x ∗ x; x2 + x = x ∗ (x + 1); x2 + 1 = (x + 1) ∗ (x + 1) from the list of four degree-2 polynomials. This leaves one prime degree-2 polynomial, namely x2 + x + 1. Next, erase all degree-3 multiples of x, x + 1, and x2 + x + 1 from the list of eight degree-3 polynomials, namely the six polynomials x3 = x ∗ x ∗ x; x3 + x2 = (x + 1) ∗ x ∗ x; x3 + x = (x + 1) ∗ (x + 1) ∗ x; x3 + x2 + x = x ∗ (x2 + x + 1); x3 + 1 = (x + 1) ∗ (x2 + x + 1); x3 + x2 + x + 1 = (x + 1) ∗ (x + 1) ∗ (x + 1). The remaining two polynomials, namely x3 + x2 + 1 and x3 + x + 1, must therefore be prime. Exercise 9. Find all prime polynomials in F2 [x] of degrees 4 and 5. [Hint: There are three prime polynomials in F2 [x] of degree 4 and six of degree 5.] Continuing in this way, we may list all prime polynomials in F2 [x] up to any desired degree. It turns out that the number N (m) of prime polynomials of F2 [x] of degree m is N (m) = 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, . . . for m = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .. (In Section 7.9 we will give a simpler method to compute N (m), and will show that N (m) > 0 for all m.) A similar sieve algorithm may be used to find the prime polynomials in F[x] over any finite field F. The algorithm starts with a listing of the monic polynomials ordered by degree, and successively erases the multiples of lower-degree prime polynomials. 7.6 A construction of a field with pm elements We now show how to construct a field with pm elements for any prime integer p and positive integer m ≥ 1. Its elements will be the set RF,m of remainder polynomials of degree less than m, and multiplication will be defined modulo an irreducible polynomial g(x) of degree m. We will subsequently show that that every finite field is isomorphic to a finite field that is constructed in this way. The construction assumes the existence of a prime polynomial g(x) ∈ Fp [x] of degree m. The proof that such a polynomial exists for all prime p and m ≥ 1 will be deferred until later. The field that we construct will be denoted by Fg(x) . The set of elements of Fg(x) will be taken to be the mod-g(x) remainder set RFp ,m = {r0 + r1 x + · · · + rm−1 xm−1 | rj ∈ Fp , 0 ≤ j ≤ m − 1}, whose size is |RFp ,m | = pm . The addition and multiplication rules will be taken to be those of mod-g(x) arithmetic. We must show that the axioms of a field are satisfied with these definitions. The associative, commutative and distributive laws for mod-g(x) arithmetic follow from the corresponding laws for ordinary polynomial arithmetic. Mod-g(x) addition of two remainder polynomials in Fg(x) yields a remainder polynomial of degree < m in Fg(x) . Fg(x) evidently forms an abelian group under mod-g(x) addition. (As already mentioned, this group is isomorphic to the additive group of (Fp )m .) Mod-g(x) multiplication of two remainder polynomials r(x), s(x) yields the remainder polynomial t(x) = r(x)s(x) mod g(x). The following exercise shows that the nonzero elements of Fg(x) form an abelian group under mod-g(x) multiplication: Exercise 10. Let g(x) be a prime polynomial of degree m, and let r(x), s(x), t(x) be polynomials in Fg(x) . (a) Prove the distributive law, i.e., (r(x)+ s(x)) ∗t(x) = r(x) ∗t(x)+ s(x) ∗t(x). [Hint: Express each product as a remainder using the Euclidean division algorithm.] (b) For r(x) =  0, show that r(x) ∗ s(x) = r(x) ∗ t(x) if s(x) =  t(x). (c) For r(x) = 0, show that as s(x) runs through all nonzero polynomials in Fg(x) , the product r(x) ∗ s(x) also runs through all nonzero polynomials in Fg(x) . (d) Using part (c) and Theorem 7.1, show that the nonzero elements of Fg(x) form an abelian group under mod-g(x) multiplication. Since we have verified the three field axioms, we have proved: Theorem 7.9 (Construction of Fg(x) ) If g(x) is an prime polynomial of degree m over a prime field Fp , then the set of remainder polynomials RFp ,m with mod-g(x) arithmetic forms a finite field Fg(x) with pm elements. Example 1. Let us construct a finite field with 22 = 4 elements using the prime degree-2 polynomial g(x) = x2 + x + 1 ∈ F2 [x]. There are four remainder polynomials mod x2 + x + 1, namely {0, 1, x, x + 1}. Addition is componentwise mod 2. For multiplication, note that x∗x = x+1 since x2 mod (x2 +x+1) = x+1. Also x ∗ x ∗ x = x ∗ (x + 1) = 1 since x3 mod (x2 + x + 1) = 1. The three nonzero elements {1, x, x + 1} thus form a cyclic group under mod-g(x) multiplication, which verifies the second field axiom for this example. The complete mod-g(x) addition and multiplication tables are as follows: ⊕ 0 1 x x+1 0 0 1 x x+1 1 1 0 x+1 x x x x+1 0 1 0 x+1 x+1 x 1 7.7 ∗ 0 1 x 1+x 1 x x+1 0 0 0 0 0 0 1 x x+1 0 x x+1 1 0 x+1 1 x The multiplicative group of F∗q is cyclic In this section we consider an arbitrary finite field Fq with q elements. By the second field axiom, the set F∗q of all q − 1 nonzero elements must form a finite abelian group under multiplication. In this section we will show that this group is actually cyclic. We start by showing that every element of F∗q is a root of the polynomial xq−1 − 1 ∈ Fq [x]. Thus we first need to discuss roots of polynomials over arbitrary fields. 7.7.1 Roots of polynomials Let F[x] be the set of polynomials over an arbitrary field F. If f (x) ∈ F[x] has a degree-1 factor x − α for some α ∈ F, then α is called a root of f (x). Since any f (x) may be uniquely expressed as f (x) = q(x)(x−α)+β for some quotient q(x) and some β ∈ F (i.e., for some remainder r(x) = β of degree less than 1), it follows that f (α) = β. Therefore α is a root of f (x) if and only if f (α) = 0 — i.e., if and only if α is a root of the polynomial equation f (x) = 0. By degree additivity, the degree of a polynomial f (x) is equal to the sum of the degrees of its prime factors, which are unique by unique factorization. Therefore a polynomial of degree m can have at most m degree-1 factors. This yields what is sometimes called the fundamental theorem of algebra: Theorem 7.10 (Fundamental theorem of algebra) Over any field F, a monic polynomial f (x) ∈ F[x] of degree m can have no more than m roots in F. If it does have m roots {β1 , . . . , βm }, then the unique factorization of f (x) is f (x) = (x − β1 ) · · · (x − βm ). Since the polynomial xn − 1 can have at most n roots in F, we have an important corollary: Theorem 7.11 (Cyclic multiplicative subgroups) In any field F, the multiplicative group F∗ of nonzero elements has at most one cyclic subgroup of any given order n. If such a subgroup exists, then its elements {1, β, . . . , β n−1 } satisfy xn − 1 = (x − 1)(x − β) · · · (x − β n−1 ). ∗ For example, the complex multiplicative group C∗ has precisely one cyclic subgroup of each finite size n, consisting of the n complex nth roots of unity. The real multiplicative group R∗ has cyclic subgroups of size 1 ({1}) and 2 ({±1}), but none of any larger size. Exercise 11. For 1 ≤ j ≤ n, the jth � �elementary symmetric function σj (S) of a set S of n elements of a field F is the sum of all nj products of j distinct elements of S. In particular, σ1 (S) is the sum of all elements of S, and σn (S) is the product of all elements of S. (a) Show that if S = {1, β, . . . , β n−1 } is a cyclic subgroup of F∗ , then σj (S) = 0 for 1 ≤ j ≤ n−1 and σn (S) = (−1)n+1 . In particular, n−1 � β j = 0, n−1 � if n > 1; j=0 β j = (−1)n+1 . j=0 Verify for S = {±1, ±i} (the four complex 4th roots of unity). (b) Prove that for any odd prime integer p, (p − 1)! = 1 · 2 · 3 · · · (p − 1) = −1 mod p. Verify for p = 3, 5 and 7. 7.7.2 Factoring xq − x over Fq For any β ∈ Fq∗ , consider the cyclic subgroup S(β) = {1, β, β 2 , β 3 , . . .} of Fq∗ generated by β. The size |S(β)| of this subgroup is called the multiplicative order of β. By the cyclic group theorem, β |S(β)| = 1, and by Lagrange’s theorem, |S(β)| must divide |F∗q | = q − 1. It follows that β q−1 = 1 for all β ∈ Fq∗ . In other words, every β ∈ Fq∗ is a root of the polynomial equation xq−1 = 1, or equivalently of the polynomial xq−1 − 1 ∈ Fq [x]. By the polynomial roots theorem, xq−1 − 1 can have at most q − 1 roots in Fq , so these are all the roots of xq−1 − 1. Thus xq−1 − 1 factors into the product of the degree-1 polynomials x − β for all β ∈ F∗q . Moreover, since 0 ∈ Fq is a root of the polynomial x and x(xq−1 − 1) = xq − x, the polynomial xq − x factors into the product of the degree-1 polynomials x − β for all β ∈ Fq . To summarize: Theorem 7.12 In a finite field Fq with q elements, every nonzero field element β ∈ Fq satisfies β q−1 = 1 and has a multiplicative order |S(β)| that divides q − 1. The nonzero elements of Fq are the q − 1 distinct roots of the polynomial xq−1 − 1 ∈ Fq [x]; i.e., � (x − β). (7.3) xq−1 − 1 = β∈F∗q The elements of Fq are the q distinct roots of the polynomial xq − x ∈ Fq [x]; i.e., � xq − x = (x − β). (7.4) β∈Fq Exercise 12. (a) Verify (7.3) for the prime field F5 . (b) Verify (7.3) for the field F4 that was constructed in Example 1. [Hint: use a symbol other than x for the indeterminate in (7.3).] 7.7.3 Every finite field has a primitive element A primitive element of a finite field Fq is an element α whose multiplicative order |S(α)| equals q − 1. If α is a primitive element, then the cyclic group {1, α, α2 , . . . , αq−2 } is a set of q − 1 distinct nonzero elements of Fq , which therefore must be all the nonzero elements. Thus if we can show that Fq has at least one primitive element, we will have shown that its nonzero elements F∗q form a cyclic group under multiplication of size q − 1. By Lagrange’s theorem, the multiplicative order |S(β)| of each nonzero element β ∈ F∗q divides q − 1. Therefore the size d of each cyclic subgroup of Fq∗ divides q − 1. As we have seen, the number of elements in a cyclic group or subgroup of size d that have order d is the Euler number φ(d). Since by the cyclic subgroups theorem F∗q has at most one cyclic subgroup of each size d, the number of elements in F∗q with order less than q − 1 is at most � φ(d). d: d|(q−1), d=q−1 But since the Euler numbers satisfy the relationship (7.1), which in this case is � q−1= φ(d), d: d|(q−1) we conclude that there must be at least φ(q − 1) elements of F∗q with order q − 1. Indeed, since F∗q has at most φ(q − 1) elements of order q − 1, all inequalities must be satisfied with equality; i.e., F∗q has precisely φ(d) elements of order d for each divisor d of q − 1. We saw in Exercise 3 that φ(q − 1) ≥ 1, so a primitive element α of order q − 1 exists. Thus is cyclic and has one cyclic subgroup of each order d that divides q − 1. This proves the following theorem: F∗q Theorem 7.13 (Primitive elements) Given any field Fq with q elements, the nonzero elements of Fq form a multiplicative cyclic group F∗q = {1, α, α2 , . . . , αq−2 }. Consequently F∗q has φ(d) ≥ 1 elements of multiplicative order d for every d that divides q − 1, and no elements of any other order. In particular, F∗q has φ(q − 1) ≥ 1 primitive elements. Henceforth we will usually write the elements of a finite field Fq as {0, 1, α, α2 , . . . , αq−2 }, where α denotes a primitive element. For Fg(x) , denoting a field element β as a power of α rather than as a remainder polynomial helps to avoid confusion when we consider polynomials in β. Example 2. The prime field F5 has φ(1) = 1 element of order 1 (the element 1), φ(2) = 1 element of order 2 (namely 4 = -1), and φ(4) = 2 primitive elements of order 4 (namely, 2 and 3). We can therefore write F5 = {0, 1, 2, 22 , 23 }, since 22 = 4 and 23 = 3 mod 5. Example 3. A field F16 = {0, 1, α, . . . , α14 } with 16 elements has • φ(1) = 1 element of order 1 (the element 1); • φ(3) = 2 elements of order 3 (α5 and α10 ); • φ(5) = 4 elements of order 5 (α3 , α6 , α9 , α12 ), and • φ(15) = 8 primitive elements of order 15 (α, α2 , α4 , α7 , α8 , α11 , α13 , α14 ). The “logarithmic” representation of the nonzero elements of Fq as distinct powers of a primitive element α is obviously highly convenient for multiplication and division. Multiplication in Fq is often carried out by using such a “log table” to convert a polynomial f (x) ∈ Fq to the exponent i such that f (x) = αi , and then using an inverse “antilog table” to convert back after adding or subtracting exponents. (Note that the zero element can be included in this scheme if we define 0 = α−∞ .) 7.8 Every finite field is isomorphic to a field Fg(x) We now wish to show that every finite field Fq is isomorphic to a field Fg(x) of the type that we have previously constructed. In particular, this will show that the number of elements of a finite field must be q = pm , a prime power. The development relies on the properties of minimal polynomials, which are the factors that appear in the unique factorization of xq − x over the prime subfield Fp of Fq . 7.8.1 Factoring xq − x into minimal polynomials over Fp Again, consider any field Fq with q elements. We have seen in Theorem 7.12 that the polynomial xq − x ∈ Fq [x] factors completely into q deqree-1 factors x − β ∈ Fq [x], β ∈ Fq . We have also seen that if Fq has characteristic p, then Fq has a prime subfield Fp with p elements. The prime subfield Fp contains the integers of Fq , which include {0, ±1}. Therefore we may regard xq − x alternatively as a polynomial in Fp [x]. By unique factorization, xq − x factors over Fp into a unique product of prime polynomials gi (x) ∈ Fp [x]: � xq − x = gi (x). (7.5) i Since each coefficient of gi (x) is an element of Fp ⊆ Fq , it is also an element of Fq , so gi (x) is also a monic polynomial in Fq [x]. We therefore have the following two factorizations of xq − x in Fq [x]: � � (7.6) (x − β) = gi (x). xq − x = i β∈Fq Since the first factorization is the unique prime factorization, it follows that each monic polynomial gi (x) of degree greater than 1 must be reducible over Fq , and must factor into a product of degree-1 monic polynomials; i.e., � deg gi (x) gi (x) = (x − βij ). (7.7) j=1 The prime polynomials gi (x) are called the minimal polynomials of Fq . Since each β ∈ Fq appears exactly once on the left side of (7.6), it also appears as a factor in exactly one minimal polynomial in (7.7). Thus the elements of Fq are partitioned into disjoint sets {βi1 , . . . , βik } where k = deg gi (x), and each β ∈ Fq is a root of exactly one minimal polynomial of Fq , called the minimal polynomial of β. The key property of the minimal polynomial of β is the following: Lemma 7.14 Let g(x) be the minimal polynomial of any given β ∈ Fq . Then g(x) is the monic polynomial of least degree in Fp [x] such that g(β) = 0. Moreover, for any f (x) ∈ Fp [x], f (β) = 0 if and only if g(x) divides f (x). Proof: Let h(x) ∈ Fp [x] be a monic polynomial of least degree such that h(β) = 0. Using the Euclidean division algorithm, g(x) = q(x)h(x) + r(x) where deg r(x) < deg h(x). Since h(β) = g(β) = 0, we must have r(β) = 0. By the smallest degree property of h(x), this implies that r(x) = 0, so h(x) divides g(x). But since g(x) is irreducible, h(x) cannot have degree less than g(x); i.e., deg h(x) = deg g(x). Moreover, since both h(x) and g(x) are monic, this implies that h(x) = g(x). Thus g(x) is the monic polynomial of least degree in Fp [x] such that g(β) = 0. Now let f (x) be any polynomial in Fp [x] that satisfies f (β) = 0. By Euclidean division, f (x) = q(x)g(x) + r(x) with deg r(x) < deg g(x). Thus r(β) = f (β) = 0. Since deg r(x) < deg g(x), r(β) = 0 if and only if r(x) = 0; i.e., if and only if g(x) divides f (x). Example 1 (cont.). Again consider the field F4 of Example 1, whose elements we now write as {0, 1, α, α2 }, where α may be taken as x or x + 1. This field has characteristic 2. The prime factorization of the binary polynomial x4 − x = x4 + x ∈ F2 [x] is x4 + x = x(x + 1)(x2 + x + 1), so the minimal polynomials of F4 are x, x + 1 and x2 + x + 1. The elements 0 and 1 ∈ F4 are the roots of x and x + 1, respectively. From (7.7), the other two elements of F4 , namely α and α2 , must be roots of x2 + x + 1 ∈ F2 [x]. We verify that x2 + x + 1 = (x + α)(x + α2 ) since α + α2 = 1 and α ∗ α2 = α3 = 1. 7.8.2 Valuation maps, minimal polynomials and subfields Given a�field Fq with prime subfield Fp , we now consider evaluating a nonzero polynomial f (x) = i fi xi ∈ Fp [x] at an element β ∈ Fq to give a value � deg f (x) f (β) = fi β i i=0 in Fq , where fi is taken as an element of Fq for the purposes of this evaluation. The value of the zero polynomial at any β is 0. The value f (β) depends on both the polynomial f (x) and the field element β ∈ Fq . Rather than regarding f (β) as a function of β, as the notation suggests, we will regard f (β) as a function of the polynomial f (x) ∈ Fp [x] for a fixed β. In other words, we consider the map mβ : Fp [x] → Fq that is defined by mβ (f (x)) = f (β). The set of values mβ (Fp [x]) of this map as f (x) ranges over polynomials in Fp [x] is by definition the subset of elements Gβ ⊆ Fq that can be expressed as linear combinations over Fp of powers of β. We will show that Gβ forms a subfield of Fq that is isomorphic to the polynomial remainder field Fg(x) , where g(x) is the minimal polynomial of β, namely the monic polynomial of least degree such that g(β) = 0. We observe that the map mβ : Fp [x] → Fq preserves addition and multiplication; i.e., mβ (f1 (x) + f2 (x)) = mβ (f1 (x)) + mβ (f2 (x)) since both sides equal f1 (β) + f2 (β), and mβ (f1 (x)f2 (x)) = mβ (f1 (x))mβ (f2 (x)) since both sides equal f1 (β)f2 (β). We can now prove the desired isomorphism between the fields Fg(x) and Gβ : Theorem 7.15 (Subfields generated by β ∈ Fq ) For any β ∈ Fq , let�g(x) be the minimal polynomial of β. Then the set of all linear combinations Gβ = {f (β) = i fi β i , f (x) ∈ Fp [x]} over Fp of powers of β is equal to the set {r(β), r(x) ∈ RFp ,m } of values of remainder polynomials r(x) ∈ RFp ,m , and Gβ is a field which is isomorphic to the field Fg(x) under the correspondence r(β) ∈ Gβ ↔ r(x) ∈ RFp ,m . Proof. We first verify that the correspondence mβ : RFp ,m → Gβ is one-to-one (invertible). First, if f (β) is any element of Gβ , then by Euclidean division we can write f (x) = q(x)g(x)+r(x) where r(x) ∈ RFp ,m , and then f (β) = q(β)g(β)+r(β) = r(β), so f (β) = r(β) for some remainder polynomial r(x). Thus mβ (RFp ,m ) = mβ (Fp [x]) = Gβ . On the other hand, no two remainder polynomials r(x), s(x) with degrees less than m can evaluate to the same element of Gβ , because if r(β) = s(β), then r(x) − s(x) is a nonzero polynomial of degree less than g(x) that evaluates to 0, contradiction. Now, as we have already seen, mβ (r(x) + s(x)) = mβ (r(x)) + mβ (s(x)) and mβ (r(x)s(x)) = mβ (r(x))mβ (s(x)), which verifies that this correspondence is an isomorphism. We remark that Gβ may be viewed as the smallest subfield of Fq containing the element β, because any subfield containing β must also contain all powers of β and all linear combinations of powers over Fp . 7.8.3 Isomorphism theorems We have shown that every finite field Fq contains a primitive element α. In this case, the subfield Gα consisting of all linear combinations over Fp of powers of α must evidently be the whole field Fq . Thus we obtain our main theorem: Theorem 7.16 (Every finite field is isomorphic to a field Fg(x) ) Every finite field Fq of characteristic p with q elements is isomorphic to a polynomial remainder field Fg(x) , where g(x) is a prime polynomial in Fp [x] of degree m. Hence q = pm for some positive integer m. Exercise 13. For which integers q, 1 ≤ q ≤ 12, does a finite field Fq exist? Finally, we wish to show that all fields with pm elements are isomorphic. The following lemma shows that every prime polynomial g(x) of degree m (we are still assuming that there exists at least one) is a minimal polynomial of every field with pm elements: m Lemma 7.17 Every prime polynomial g(x) ∈ Fp [x] of degree m divides xp − x. Proof. If g(x) is a prime polynomial in Fp [x] of degree m, then the set RFp ,m with mod-g(x) arithmetic forms a field Fg(x) with pm elements. The remainder polynomial x ∈ RFp ,m is a field element β ∈ Fg(x) . Evidently g(β) = 0, but r(β) = 0 if deg r(x) < m; therefore g(x) is the m m minimal polynomial of β. Since β p −1 = 1, β is a root of xp −1 − 1. This implies that g(x) m m divides xp −1 − 1, and thus also xp − x. Consequently every field of size pm includes m elements whose minimal polynomial is g(x). Therefore by the same construction as above, we can prove: Theorem 7.18 (All finite fields of the same size are isomorphic) For any prime polynomial g(x) ∈ Fp [x] of degree m, every field of pm elements is isomorphic to the polynomial remainder field Fg(x) . m More on the factorization of xp − x 7.8.4 We can now obtain further information on the factorization of xq − x. In view of Theorem 7.16, we now set q = pm . We first show that the set of roots of a minimal polynomial gi (x) ∈ Fp [x] is closed under the operation of taking the pth power. This follows from the curious but important fact that over a field F of characteristic p, taking the pth power is a linear operation. For example, when p = 2, squaring is linear because (α + β)2 = α2 + αβ + αβ + β 2 = α2 + β 2 . More generally, over any field F, p (α + β) = p � � � p j=0 j αj β p−j , �� �� where pj αj β p−j denotes the sum of pj terms equal to αj β p−j . If F has characteristic p, then �� the integer pj = p!/(j!)((n − j)!) may be reduced mod p. Now p! contains a factor of p, but �� for 1 ≤ j ≤ p − 1, j! and (n − j)! do not contain a factor of p. Therefore pj = 0 mod p for 1 ≤ j ≤ p − 1, and (α + β)p = αp + β p . By taking the pth power n times, we may extend this result as follows: Lemma 7.19 (Linearity of taking the pn th power) Over any field F of characteristic p, for any n ≥ 1, taking the pn th power is linear; i.e., n n n (α + β)p = αp + β p . m Note that if F has q = pm elements, then β p repetitive for n ≥ m. = β for all β ∈ F, so this lemma becomes Exercise 14. Using this lemma, prove that if f (x) = n n n �m i i=0 fi x , n n then n n n n p mp f p (x) = (f0 + f1 x + f2 x2 + · · · + fm xm )p = f0p + f1p xp + f2p x2p + · · · + fm x . This result yields a useful test for whether a polynomial f (x) ∈ F[x] is in Fp [x] or not, and a useful formula in case it is: Lemma 7.20 (Prime subfield polynomials) For any field F of characteristic p and any f (x) ∈ F[x], f p (x) = f (xp ) if and only if f (x) ∈ Fp [x]; i.e., if and only if all coefficients fi are in the prime subfield Fp ⊆ F. Proof. By Exercise 14, we have f p (x) = (f0 + f1 x + f2 x2 + · · · + fn xn )p = f0p + f1p xp + f2p x2p + · · · + fnp xnp . Now the elements of F that are in Fp are precisely the p roots of the polynomial xp − x; thus β p = β if and only if β ∈ Fp . Thus the right side of this equation simplifies to f (xp ) if and only if fi ∈ Fp for all i. Exercise 15. Prove that a positive integer n is prime if and only if (x − a)n = xn − a mod n for every integer a that is relatively prime to n.5 Using Lemma 7.20, we now show that the roots of a minimal polynomial are a cyclotomic coset 2 of the form {β, β p , β p , . . .}: Theorem 7.21 (Roots of minimal polynomials) Let g(x) be a minimal polynomial of a fi2 n−1 nite field F with pm elements. Then the roots of g(x) are a set of the form {β, β p , β p , . . . , β p }, n where n is a divisor of m. Moreover, g(x) divides xp − x. Proof. Let β be any root of g(x). Since g(x) ∈ Fp [x], Lemma 7.20 shows that g(xp ) = g p (x). 2 3 i Therefore g(β p ) = g p (β) = 0. Thus β p is also a root of g(x). Iterating, β p , β p , . . . , β p , . . . are all roots of g(x). Because F is finite, these roots cannot all be distinct. Therefore let n be the n j j j+k smallest integer such that β p = β. Thus β p = β for 1 ≤ j < n. This implies that β p = β p 2 n−1 for 0 ≤ j < n, 1 ≤ k < n; i.e., all elements of the set {β, β p , β p , . . . , β p } are distinct. Thus 2 m j β, β p , β p , . . . is a cyclic sequence and β p = β if and only if n is a divisor of j. Since β p = β, we see that n must divide m. Finally, we show that these roots are all of the roots of g(x); i.e., deg g(x) = n and g(x) = n−1 � i (x − β p ). i=0 The right side of this equation is a monic polynomial h(x) ∈ F[x] of degree n. Since the roots of h(x) are roots of g(x), h(x) must divide g(x) in F[x]. Now, using Lemma 7.20, we can prove that h(x) is actually a polynomial in Fp [x], because p h (x) = n−1 � i=0 pi p (x − β ) = n−1 � (x − β p i=0 pi+1 )= n−1 � i (xp − β p ) = h(xp ), i=0 n where we use the linearity of taking the pth power and the fact that β p = β. Therefore, since g(x) has no factors in Fp [x], g(x) must actually be equal to h(x). n n Finally, since the roots of g(x) all satisfy β p = β, they are all roots of the polynomial xp − x, n which implies that g(x) divides xp − x. 5 This is the basis of the polynomial-time primality test of [Agrawal, Kayal and Saxena, 2002]. This theorem has some important implications. First, the degree n of a minimal polynomial g(x) of a finite field F with pm elements must be a divisor of m. Second, the subfield Gβ of F n m generated by a root β of g(x) must have pn elements. Third, xp − x divides xp − x, since the n m elements of Gβ are all the roots of xp − x and are also roots of xp − x. Conversely, let g(x) be any prime polynomial in Fp [x] of degree n. Then there is a finite field n generated by g(x) with pn elements. This proves that g(x) divides xp − x, and thus g(x) divides m m xp − x for every multiple m of n. Thus the divisors of xp − x include every prime polynomial in Fp [x] whose degree n divides m. m Moreover, xp − x has no repeated factors. We proved this earlier assuming the existence of a field F with pm elements; however, we desire a proof that does not make this assumption. The following exercise yields such a proof. m Exercise 16 (xp −x has no repeated factors). The formal derivative of a degree-n polynomial f (x) ∈ Fp [x] is defined as n �  f (x) = (j mod p)fj xj−1 j=1 (a) Show that if f (x) = g(x)h(x), then f  (x) = g  (x)h(x) + g(x)h (x). (b) Show that an prime polynomial g(x) is a repeated divisor of f (x) if and only if g(x) is a divisor of both f (x) and f  (x). m (c) Show that xp − x has no repeated prime factors over Fp . m Now we can conclude our discussion of the factorization of xp − x as follows: m m Theorem 7.22 (Factors of xp − x) The polynomial xp − x factors over Fp into the product of the prime polynomials in Fp [x] whose degrees divide m, with no repetitions. For example, over F2 , we have x2 + x = x(x + 1); x4 + x = x(x + 1)(x2 + x + 1); x8 + x = x(x + 1)(x3 + x2 + 1)(x3 + x + 1); x16 + x = x(x + 1)(x2 + x + 1)(x4 + x3 + 1)(x4 + x3 + x2 + x + 1)(x4 + x + 1). Exercise 17. Find all prime polynomials g(x) ∈ F3 [x] of degree 1 and 2 over the ternary field F3 . Show that the product of these polynomials is x9 − x = x9 + 2x. Explain, with reference to F9 . 7.9. FINITE FIELDS FP M EXIST FOR ALL PRIME P AND M ≥ 1 7.9 Finite fields Fpm exist for all prime p and m ≥ 1 At last we can prove that for every prime p and positive integer m there exists a prime polynomial g(x) ∈ Fp [x] of degree m. This will prove the existence of a finite field Fg(x) with pm elements. Using the factorization of Theorem 7.22, we will show that there do not exist enough prime polynomials of degree less than m that their product could have degree pm . Let N (n) denote the number of prime polynomials over Fp of degree n. The product of these m polynomials has degree nN (n). Since xp − x is the product of these polynomials for all divisors n of m, and there are no repeated factors, its degree pm is equal to � pm = nN (n) (7.8) n: n|m This formula may be solved recursively for each N (m), starting with N (1) = p. Exercise 18. Calculate N (m) for p = 2 for m = 1 to 10. Check your results against those stated in Section 7.5.4. Now we are in a position to prove the desired theorem: Theorem 7.23 (Existence of prime polynomials) Let N (m) be the number of prime polynomials in Fp [x] of degree m, which is given recursively by (7.8). For every prime p and positive integer m, N (m) > 0. Proof. Note first that nN (n) ≤ pn . Thus � pm ≤ mN (m) + pn ≤ mN (m) + (m/2)pm/2 , n