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Encyclopedia > Finite field

In 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, as will be described below. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Galois at the age of fifteen from the pencil of a classmate. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, Galois theory is a branch of abstract algebra. ... The German Lorenz cipher machine Cryptography or cryptology is a field of mathematics and computer science concerned with information security and related issues, particularly encryption and authentication. ... Coding theory is a branch of mathematics and computer science dealing with the error-prone process of transmitting data across noisy channels, via clever means, so that a large number of errors that occur can be corrected. ...

Contents


Classification

The finite fields are classified as follows[1]:

  • Every finite field has pn elements for some prime p and some integer n ≥ 1. (This p is the characteristic of the field.)
  • For every prime p and integer n ≥ 1, there exists a finite field with pn elements.
  • All fields with pn elements are isomorphic (that is, their addition tables are essentially the same, and their multiplication tables are essentially the same). This justifies using the same name for all of them; they are denoted by GF(pn). The letters "GF" stand for "Galois field".

For example, there is 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. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...


The simplest case is when n = 1. In this case the finite field GF(p) is the ring Z/pZ. This 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. Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... The p-adic number systems were first described by Kurt Hensel in 1897. ...


Even though GF(p) = Z/pZ, the finite field GF(pn) should not be confused with Z/pnZ (the 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. This article is about the mathematical group. ...


Proof of the classification

Suppose that F is a finite field. The characteristic of F cannot be zero, because then F would contain infinitely many elements generated by the multiplicative identity: 0, 1, 1+1, 1+1+1, etc; so would be infinite. Therefore, since the characteristic of a field is either 0 or prime, the characteristic is a prime number p. It contains a subfield Z/pZ consisting of the p (distinct) elements 0, 1, 2, ..., p−1, where for example 2 means 1+1. Furthermore, 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. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In mathematics, the dimension of a vector space V is the cardinality (i. ...


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) = TqT, 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: In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X − ai, and such that the ai generate L over K. It can be shown that such splitting fields exist...

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 In mathematics, a splitting field of a polynomial with coefficients in a field is an extension of that field over which the polynomial factors into linear factors. ... This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ... In mathematics, the formal derivative is an operation on elements of a polynomial ring which mimics the form of the derivative from calculus. ...

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

(r + s)q = rq + sq = r + s,

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 In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

(rs)q = rqsq = rs

and

(r − 1)q = (rq) − 1 = r − 1.

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 fields

Given 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). A prime power is a positive integer power of a prime. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In mathematics, the minimal polynomial of an object α is the monic polynomial p of least degree such that p(α)=0. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...


Examples

The 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 facts

If F is a finite field with q = pn elements (where p is prime), then

xq = x

for all x in F. Furthermore, the map

f : FF

defined by

f(x) = xp

is bijective and a homomorphism, and is therefore an automorphism. It is called the Frobenius automorphism, after Ferdinand Georg Frobenius. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In abstract algebra, a homomorphism is a structure-preserving map. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, the Frobenius automorphism is an automorphism induced by a prime power mapping defined for various extensions of fields. ... Picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ...


The Frobenius automorphism has order n, so that the cyclic group it generates is the full group of automorphisms of the field. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...


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). In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...


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. In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...


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. In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... Joseph Henry Maclagen Wedderburn (2 February 1882- 9 October 1948) was a Scottish mathematician, who from 1909 had positions at Princeton University. ...


Applications

The 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 In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

F = { 0, 1, x, x2, ..., xq-2 }.

Unless q = 2 or 3, the element x is not unique. If we fix one, then for any non-zero element a in Fq, there is a unique integer n with

0 ≤ nq − 2

such that

a = xn.

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. In abstract algebra and its applications, the discrete logarithms are defined in group theory in analogy to ordinary logarithms. ... The German Lorenz cipher machine Cryptography or cryptology is a field of mathematics and computer science concerned with information security and related issues, particularly encryption and authentication. ...


Finite fields also find applications in coding theory: many codes are constructed as subspaces of vector spaces over finite fields. Coding theory is a branch of mathematics and computer science dealing with the error-prone process of transmitting data across noisy channels, via clever means, so that a large number of errors that occur can be corrected. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ...


Some small finite fields

GF(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 also

Arithmetic in a finite field is different from standard arithmetic. ... In mathematics, a quasi-finite field is a generalisation of a finite field. ...

References

  1. ^ p287, Jacobson, Nathan (1985). Basic Algebra I, 2nd Ed., New York: W. H. Freeman and Company. ISBN 0716714809.

  Results from FactBites:
 
Finite field - Wikipedia, the free encyclopedia (1272 words)
Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory.
The multiplicative group of every finite field is cyclic, a special case of a theorem mentioned here in the article about fields.
Finite fields also find applications in coding theory: many codes are constructed as subspaces of vector spaces over finite fields.
Finite field - definition of Finite field in Encyclopedia (684 words)
Since every field of characteristic 0 contains the rationals and is therefore infinite, all finite fields have prime characteristic.
The multiplicative group of every finite field is cyclic, a special case of a theorem mentioned in the article about fields.
The value of n for a given a is called the discrete log of a (in the given field, to base x).
  More results at FactBites »

 
 

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