Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number.

Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory.

A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms.

The number of elements of a finite field is called its order or, sometimes, its size. A finite field of order q exists if and only if q is a prime power pk (where p is a prime number and k is a positive integer). In a field of order pk, adding p copies of any element always results in zero; that is, the characteristic of the field is p.

In a finite field of order q, the polynomial XqX has all q elements of the finite field as roots. The non-zero elements of a finite field form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powers of a single element called a primitive element of the field. (In general there will be several primitive elements for a given field.)

The elements of the prime field of order p may be represented by integers in the range 0, ..., p − 1. The sum, the difference and the product are the remainder of the division by p of the result of the corresponding integer operation. The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see Extended Euclidean algorithm § Modular integers).

(sometimes called the freshman's dream) is true in a field of characteristic p. This follows from the binomial theorem, as each binomial coefficient of the expansion of (x + y)p, except the first and the last, is a multiple of p.

By Fermat's little theorem, if p is a prime number and x is in the field GF(p) then xp = x. This implies the equality

for polynomials over GF(p). More generally, every element in GF(pn) satisfies the polynomial equation xpnx = 0.

Any finite field extension of a finite field is separable and simple. That is, if E is a finite field and F is a subfield of E, then E is obtained from F by adjoining a single element whose minimal polynomial is separable. To use a jargon, finite fields are perfect.

A more general algebraic structure that satisfies all the other axioms of a field, but whose multiplication is not required to be commutative, is called a division ring (or sometimes skew field). By Wedderburn's little theorem, any finite division ring is commutative, and hence is a finite field.

Let q = pn be a prime power, and F be the splitting field of the polynomial

over the prime field GF(p). This means that F is a finite field of lowest order, in which P has q distinct roots (the formal derivative of P is P = −1, implying that gcd(P, P) = 1, which in general implies that the splitting field is a separable extension of the original). The above identity shows that the sum and the product of two roots of P are roots of P, as well as the multiplicative inverse of a root of P. In other words, the roots of P form a field of order q, which is equal to F by the minimality of the splitting field.

The uniqueness up to isomorphism of splitting fields implies thus that all fields of order q are isomorphic. Also, if a field F has a field of order q = pk as a subfield, its elements are the q roots of XqX, and F cannot contain another subfield of order q.

In summary, we have the following classification theorem first proved in 1893 by E. H. Moore:[1]

It follows that GF(pn) contains a subfield isomorphic to GF(pm) if and only if m is a divisor of n; in that case, this subfield is unique. In fact, the polynomial XpmX divides XpnX if and only if m is a divisor of n.

Given a prime power q = pn with p prime and n > 1, the field GF(q) may be explicitly constructed in the following way. One first chooses an irreducible polynomial P in GF(p)[X] of degree n (such an irreducible polynomial always exists). Then the quotient ring

of the polynomial ring GF(p)[X] by the ideal generated by P is a field of order q.

More explicitly, the elements of GF(q) are the polynomials over GF(p) whose degree is strictly less than n. The addition and the subtraction are those of polynomials over GF(p). The product of two elements is the remainder of the Euclidean division by P of the product in GF(p)[X]. The multiplicative inverse of a non-zero element may be computed with the extended Euclidean algorithm; see .

Except in the construction of GF(4), there are several possible choices for P, which produce isomorphic results. To simplify the Euclidean division, for P one commonly chooses polynomials of the form

which make the needed Euclidean divisions very efficient. However, for some fields, typically in characteristic 2, irreducible polynomials of the form Xn + aX + b may not exist. In characteristic 2, if the polynomial Xn + X + 1 is reducible, it is recommended to choose Xn + Xk + 1 with the lowest possible k that makes the polynomial irreducible. If all these trinomials are reducible, one chooses "pentanomials" Xn + Xa + Xb + Xc + 1, as polynomials of degree greater than 1, with an even number of terms, are never irreducible in characteristic 2, having 1 as a root.[3]

A possible choice for such a polynomial is given by Conway polynomials. They ensure a certain compatibility between the representation of a field and the representations of its subfields.

In the next sections, we will show how the general construction method outlined above works for small finite fields.

This may be deduced as follows from the results of the preceding section.

Therefore, for GF(4) the construction of the preceding section must involve this polynomial, and

and that α and 1 + α are the elements of GF(4) that are not in GF(2). The tables of the operations in GF(4) result from this, and are as follows:

A table for subtraction is not given, because subtraction is identical to addition, as is the case for every field of characteristic 2. In the third table, for the division of x by y, the values of x must be read in the left column, and the values of y in the top row. (Because 0 ⋅ z = 0 for every z in every ring the division by 0 has to remain undefined.)

For applying the above general construction of finite fields in the case of GF(p2), one has to find an irreducible polynomial of degree 2. For p = 2, this has been done in the preceding section. If p is an odd prime, there are always irreducible polynomials of the form X2r, with r in GF(p).

More precisely, the polynomial X2r is irreducible over GF(p) if and only if r is a quadratic non-residue modulo p (this is almost the definition of a quadratic non-residue). There are p − 1/2 quadratic non-residues modulo p. For example, 2 is a quadratic non-residue for p = 3, 5, 11, 13, ..., and 3 is a quadratic non-residue for p = 5, 7, 17, .... If p ≡ 3 mod 4, that is p = 3, 7, 11, 19, ..., one may choose −1 ≡ p − 1 as a quadratic non-residue, which allows us to have a very simple irreducible polynomial X2 + 1.

Having chosen a quadratic non-residue r, let α be a symbolic square root of r, that is a symbol which has the property α2 = r, in the same way as the complex number i is a symbolic square root of −1. Then, the elements of GF(p2) are all the linear expressions

with a and b in GF(p). The operations on GF(p2) are defined as follows (the operations between elements of GF(p) represented by Latin letters are the operations in GF(p)):

is irreducible over GF(2) and GF(3), that is, it is irreducible modulo 2 and 3 (to show this, it suffices to show that it has no root in GF(2) nor in GF(3)). It follows that the elements of GF(8) and GF(27) may be represented by expressions

The addition, additive inverse and multiplication on GF(8) and GF(27) may thus be defined as follows; in following formulas, the operations between elements of GF(2) or GF(3), represented by Latin letters, are the operations in GF(2) or GF(3), respectively:

is irreducible over GF(2), that is, it is irreducible modulo 2. It follows that the elements of GF(16) may be represented by expressions

where a, b, c, d are either 0 or 1 (elements of GF(2)), and α is a symbol such that

(that is, α is defined as a root of the given irreducible polynomial). As the characteristic of GF(2) is 2, each element is its additive inverse in GF(16). The addition and multiplication on GF(16) may be defined as follows; in following formulas, the operations between elements of GF(2), represented by Latin letters are the operations in GF(2).

The set of non-zero elements in GF(q) is an abelian group under the multiplication, of order q – 1. By Lagrange's theorem, there exists a divisor k of q – 1 such that xk = 1 for every non-zero x in GF(q). As the equation xk = 1 has at most k solutions in any field, q – 1 is the lowest possible value for k. The structure theorem of finite abelian groups implies that this multiplicative group is cyclic, that is, all non-zero elements are powers of a single element. In summary:

Such an element a is called a primitive element. Unless q = 2, 3, the primitive element is not unique. The number of primitive elements is φ(q − 1) where φ is Euler's totient function.

The result above implies that xq = x for every x in GF(q). The particular case where q is prime is Fermat's little theorem.

If a is a primitive element in GF(q), then for any non-zero element x in F, there is a unique integer n with 0 ≤ nq − 2 such that

While an can be computed very quickly, for example using exponentiation by squaring, there is no known efficient algorithm for computing the inverse operation, the discrete logarithm. This has been used in various cryptographic protocols, see Discrete logarithm for details.

When the nonzero elements of GF(q) are represented by their discrete logarithms, multiplication and division are easy, as they reduce to addition and subtraction modulo q – 1. However, addition amounts to computing the discrete logarithm of am + an. The identity

allows one to solve this problem by constructing the table of the discrete logarithms of an + 1, called Zech's logarithms, for n = 0, ..., q − 2 (it is convenient to define the discrete logarithm of zero as being −∞).

Zech's logarithms are useful for large computations, such as linear algebra over medium-sized fields, that is, fields that are sufficiently large for making natural algorithms inefficient, but not too large, as one has to pre-compute a table of the same size as the order of the field.

Every nonzero element of a finite field is a root of unity, as xq−1 = 1 for every nonzero element of GF(q).

If n is a positive integer, an nth primitive root of unity is a solution of the equation xn = 1 that is not a solution of the equation xm = 1 for any positive integer m < n. If a is a nth primitive root of unity in a field F, then F contains all the n roots of unity, which are 1, a, a2, ..., an−1.

The field GF(q) contains a nth primitive root of unity if and only if n is a divisor of q − 1; if n is a divisor of q − 1, then the number of primitive nth roots of unity in GF(q) is φ(n) (Euler's totient function). The number of nth roots of unity in GF(q) is gcd(n, q − 1).

In a field of characteristic p, every (np)th root of unity is also a nth root of unity. It follows that primitive (np)th roots of unity never exist in a field of characteristic p.

The field GF(64) has several interesting properties that smaller fields do not share: it has two subfields such that neither is contained in the other; not all generators (elements with minimal polynomial of degree 6 over GF(2)) are primitive elements; and the primitive elements are not all conjugate under the Galois group.

The order of this field being 26, and the divisors of 6 being 1, 2, 3, 6, the subfields of GF(64) are GF(2), GF(22) = GF(4), GF(23) = GF(8), and GF(64) itself. As 2 and 3 are coprime, the intersection of GF(4) and GF(8) in GF(64) is the prime field GF(2).

The union of GF(4) and GF(8) has thus 10 elements. The remaining 54 elements of GF(64) generate GF(64) in the sense that no other subfield contains any of them. It follows that they are roots of irreducible polynomials of degree 6 over GF(2). This implies that, over GF(2), there are exactly 9 = 54/6 irreducible monic polynomials of degree 6. This may be verified by factoring X64X over GF(2).

They form two orbits under the action of the Galois group. As the two factors are reciprocal to each other, a root and its (multiplicative) inverse do not belong to the same orbit.
They split into 6 orbits of 6 elements under the action of the Galois group.

This shows that the best choice to construct GF(64) is to define it as GF(2)[X] / (X6 + X + 1). In fact, this generator is a primitive element, and this polynomial is the irreducible polynomial that produces the easiest Euclidean division.

is a GF(p)-linear endomorphism and a field automorphism of GF(q), which fixes every element of the subfield GF(p). It is called the Frobenius automorphism, after Ferdinand Georg Frobenius.

It has been shown in the preceding section that φn is the identity. For 0 < k < n, the automorphism φk is not the identity, as, otherwise, the polynomial

There are no other GF(p)-automorphisms of GF(q). In other words, GF(pn) has exactly n GF(p)-automorphisms, which are

In terms of Galois theory, this means that GF(pn) is a Galois extension of GF(p), which has a cyclic Galois group.

The fact that the Frobenius map is surjective implies that every finite field is perfect.

If F is a finite field, a non-constant monic polynomial with coefficients in F is irreducible over F, if it is not the product of two non-constant monic polynomials, with coefficients in F.

As every polynomial ring over a field is a unique factorization domain, every monic polynomial over a finite field may be factored in a unique way (up to the order of the factors) into a product of irreducible monic polynomials.

There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite field. They are a key step for factoring polynomials over the integers or the rational numbers. At least for this reason, every computer algebra system has functions for factoring polynomials over finite fields, or, at least, over finite prime fields.

factors into linear factors over a field of order q. More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order q.

This implies that, if q = pn then XqX is the product of all monic irreducible polynomials over GF(p), whose degree divides n. In fact, if P is an irreducible factor over GF(p) of XqX, its degree divides n, as its splitting field is contained in GF(pn). Conversely, if P is an irreducible monic polynomial over GF(p) of degree d dividing n, it defines a field extension of degree d, which is contained in GF(pn), and all roots of P belong to GF(pn), and are roots of XqX; thus P divides XqX. As XqX does not have any multiple factor, it is thus the product of all the irreducible monic polynomials that divide it.

This property is used to compute the product of the irreducible factors of each degree of polynomials over GF(p); see Distinct degree factorization.

Number of monic irreducible polynomials of a given degree over a finite field

The number N(q, n) of monic irreducible polynomials of degree n over GF(q) is given by[4]

where μ is the Möbius function. This formula is almost a direct consequence of above property of XqX.

By the above formula, the number of irreducible (not necessarily monic) polynomials of degree n over GF(q) is (q − 1)N(q, n).

One may easily deduce that, for every q and every n, there is at least one irreducible polynomial of degree n over GF(q). This lower bound is sharp for q = n = 2.

In cryptography, the difficulty of the discrete logarithm problem in finite fields or in elliptic curves is the basis of several widely used protocols, such as the Diffie–Hellman protocol. For example, in 2014, a secure internet connection to Wikipedia involved the elliptic curve Diffie–Hellman protocol (ECDHE) over a large finite field.[5] In coding theory, many codes are constructed as subspaces of vector spaces over finite fields.

Finite fields are widely used in number theory, as many problems over the integers may be solved by reducing them modulo one or several prime numbers. For example, the fastest known algorithms for polynomial factorization and linear algebra over the field of rational numbers proceed by reduction modulo one or several primes, and then reconstruction of the solution by using Chinese remainder theorem, Hensel lifting or the LLL algorithm.

Similarly many theoretical problems in number theory can be solved by considering their reductions modulo some or all prime numbers. See, for example, Hasse principle. Many recent developments of algebraic geometry were motivated by the need to enlarge the power of these modular methods. Wiles' proof of Fermat's Last Theorem is an example of a deep result involving many mathematical tools, including finite fields.

The Weil conjectures concern the number of points on algebraic varieties over finite fields and the theory has many applications including exponential and character sum estimates.

Finite fields have widespread application in combinatorics, two well known examples being the definition of Paley Graphs and the related construction for Hadamard Matrices. In arithmetic combinatorics finite fields[6] and finite field models[7][8] are used extensively, such as in Szemerédi's theorem on arithmetic progressions.

Although finite fields are not algebraically closed, they are quasi-algebraically closed, which means that every homogeneous polynomial over a finite field has a non-trivial zero whose components are in the field if the number of its variables is more than its degree. This was a conjecture of Artin and Dickson proved by Chevalley (see Chevalley–Warning theorem).

A division ring is a generalization of field. Division rings are not assumed to be commutative. There are no non-commutative finite division rings: Wedderburn's little theorem states that all finite division rings are commutative, hence finite fields. The result holds even if we relax associativity and consider alternative rings, by the Artin–Zorn theorem.[9]