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Finite fieldFrom CryptoDox, The Online Encyclopedia on Cryptography and Information SecurityIn abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory. The finite fields are completely known.
ClassificationThe finite fields are classified as follows Template:Harv:
This classification justifies using a naming scheme for finite fields that specifies only the order of the field. One notation for a finite field is GF(pn), where the letters "GF" stand for "Galois field". Another common notation is ExamplesThere exists a finite field GF(4) = GF(22) with 4 elements, and every field with 4 elements is isomorphic to this one. There is also a finite field GF(8) = GF(23) with 8 elements, and every field with 8 elements is isomorphic to this one. However, there is no finite field with 6 elements, because 6 is not a power of any prime. The simplest case is when the order of the field is prime, i.e., n = 1. This finite field, GF(p), is the ring Z/pZ. It is a finite field with p elements, usually labelled 0, 1, 2, ... p−1, where arithmetic is performed modulo p. It is also sometimes denoted by Zp, but this may cause confusion because the same notation is used for the ring of p-adic integers. Even though GF(p) = Z/pZ, the finite field GF(pn) must not be confused with Z/pnZ (the ring of integers modulo pn) for n ≥ 2. In fact, for n ≥ 2, the latter is not even a field; for example GF(4) is not the same thing as Z/4Z (the integers modulo 4). Rather, the underlying additive group of GF(4) is isomorphic to the Klein four-group, (Z/2Z)2. Proof of the classificationSuppose that F is a finite field with additive identity 0 and multiplicative identity 1. The characteristic of F is a prime number p as it is the characteristic of a finite ring without zero divisors. The p (distinct) elements 0, 1, 2, ..., p−1 (where for example 2 means 1+1) form a subfield of F that is isomorphic to Z/pZ. F is a vector space over Z/pZ, and it must have finite dimension over Z/pZ. If the dimension is n, then each element of F is specified uniquely by n coordinates in Z/pZ. There are p possibilities for each coordinate, so the number of elements in F is pn. This proves the first statement. The proof of the second statement, concerning the existence of a finite field for a given p and n, is more involved. Consider the polynomial f(T) = Tq − T, where q = pn. It is possible to construct a field F (called the splitting field of f), which contains Z/pZ, and which is large enough for f(T) to split completely into linear factors: where each ri is an element of F. (The existence of splitting fields in general is discussed in construction of splitting fields.) These q roots are distinct, because f(T) is a polynomial of degree q, and it has no repeated roots because its derivative is which has no roots in common with f(T). Furthermore, setting R to be the set of these roots, one sees that R itself forms a field, as follows. Both 0 and 1 are in R, because 0q = 0 and 1q = 1. If r and s are in R, then so that r+s is in R; the first equality above follows from the binomial theorem and the fact that F has characteristic p. Therefore R is closed under addition. Similarly, R is closed under multiplication and taking inverses, because and Therefore R is a field with q elements, proving the second statement. Finally the uniqueness statement: R is itself the splitting field of f(T), because it is generated over Z/pZ by the roots of f(T), and the splitting field of any polynomial is unique up to isomorphism. Explicitly constructing finite fieldsGiven a prime power q = pn, we may explicitly construct GF(q), the finite field with q elements, as follows. Select an irreducible polynomial f(T) of degree n with coefficients in GF(p). (Such an f is guaranteed to exist, once we know that the finite field GF(q) exists: just take the minimal polynomial of any element that generates GF(q) over the subfield GF(p).) Then GF(q) = GF(p)[T] / <f(T)>. Here, GF(p)[T] denotes the ring of all polynomials with coefficients in GF(p), <f(T)> denotes the ideal generated by f(T), and the quotient is meant in the sense of quotient rings — the set of polynomials with coefficients in GF(p) on division by f(T). ExamplesThe polynomial f(T) = T 2 + T + 1 is irreducible over GF(2), and GF(4) = GF(2)[T] / <T2+T+1> can therefore be written as the set {0, 1, t, t+1} where the multiplication is carried out by using the relation t2 + t + 1 = 0. In fact, since we are working over GF(2) (that is, over characteristic 2), we may write this as t2 = t + 1. (This follows because −1 = 1 in GF(2)!) Then, for example, to determine t3, we calculate: t3 = t(t2) = t(t+1) = t2+t = t+1+t = 2t + 1 = 1, so t3 = 1. In order to find the multiplicative inverse of t in this field, we have to find a polynomial p(T) such that T * p(T) = 1 modulo T 2 + T + 1. The polynomial p(T) = T + 1 works, and hence 1/t = t + 1. To construct the field GF(27), we could start for example with the irreducible polynomial T 3 + T 2 + T + 2 over GF(3). We then have GF(27) = {at2 + bt + c : a, b, c in GF(3)}, where the multiplication is defined by t 3 + t 2 + t + 2 = 0, or by rearranging this equation, t3 = 2t2 + 2t + 1. Properties and factsIf F is a finite field with q = pn elements (where p is prime), then
for all x in F. Furthermore, the map
defined by
is bijective and a homomorphism, and is therefore an automorphism. It is called the Frobenius automorphism, after Ferdinand Georg Frobenius. The Frobenius automorphism has order n, thus the cyclic group it generates is the full group of automorphisms of the field. The field GF(pm) contains a copy of GF(pn) if and only if n divides m. The reason for the if direction is that there exist irreducible polynomials of every degree over GF(pm). If we actually construct our finite fields in such a fashion that GF(pn) is contained in GF(pm) whenever n divides m, then we can form the union of all these fields. This union is also a field, albeit an infinite one. It is the algebraic closure of each of the fields GF(pn). Even if we don't construct our fields this way, we can still speak of the algebraic closure, but some more delicacy is required in its construction. For all fields multiplication is commutative. For finite fields however commutativity of multiplication can be shown to follow from the other properties of a field. Division rings are algebraic structures more general than fields, as they are not assumed to be necessarily commutative. Wedderburn's (little) theorem states that all finite division rings are commutative, hence finite fields. ApplicationsThe multiplicative group of every finite field is cyclic, a special case of a theorem mentioned here in the article about fields. This means that if F is a finite field with q elements, then there always exists an element x in F such that
Unless q = 2 or 3, the element x is not unique. In particular, the set of such elements x has size Failed to parse (unknown error\scriptstyle): \scriptstyle\varphi(q-1) where phi is euler's totient function. If we fix one, then for any non-zero element a in Fq, there is a unique integer n with
such that
The value of n for a given a is called the discrete log of a (in the given field, to base x). In practice, although calculating xn is relatively trivial given n, finding n for a given a is (under current theories) a computationally difficult process, and has many applications in cryptography. Finite fields also find applications in coding theory: many codes are constructed as subspaces of vector spaces over finite fields. Some small finite fieldsGF(2): + | 0 1 · | 0 1 --+---- --+---- 0 | 0 1 0 | 0 0 1 | 1 0 1 | 0 1 GF(3): + | 0 1 2 · | 0 1 2 --+------ --+------ 0 | 0 1 2 0 | 0 0 0 1 | 1 2 0 1 | 0 1 2 2 | 2 0 1 2 | 0 2 1 GF(4): + | 0 1 A B · | 0 1 A B --+-------- --+-------- 0 | 0 1 A B 0 | 0 0 0 0 1 | 1 0 B A 1 | 0 1 A B A | A B 0 1 A | 0 A B 1 B | B A 1 0 B | 0 B 1 A See alsoReferencesExternal links
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