Finite Field

A finite field is a Field with a finite Order (number of elements), also called aGalois Field. The order of a finite field is always a Prime or a Power of a Prime (Birkhoff and MacLane 1965). For each Prime Power, there exists exactly one (up to an Isomorphism) finite field GF(),often written as in current usage. GF() is called the Prime Field of order , and is theField of Residue Classes modulo , where the elements are denoted 0, 1, ..., . in GF() means the same as . Note, however, that in the Ring ofresidues modulo 4, so 2 has no reciprocal, and the Ring of residues modulo 4 is distinct from the finite field withfour elements. Finite fields are therefore denoted GF(), instead of GF() for clarity.

The finite field GF(2) consists of elements 0 and 1 which satisfy the following addition and multiplication tables.

	0	1
0	0	1
1	1	0

	0	1
0	0	0
1	0	1

If a subset of the elements of a finite field satisfies the above Axioms with the same operatorsof , then is called a Subfield. Finite fields are used extensively in the study of Error-CorrectingCodes.

When , GF() can be represented as the Field of Equivalence Classes ofPolynomials whose Coefficients belong to GF(). Any IrreduciblePolynomial of degree yields the same Field up to an Isomorphism. For example, for GF(), themodulus can be taken as , , or any other Irreducible Polynomial of degree 3. Using the modulus, the elements of GF()--written 0, , , ...--can be represented asPolynomials with degree less than 3. For instance,

Now consider the following table which contains several different representations of the elements of a finite field. Thecolumns are the power, polynomial representation, triples of polynomial representation Coefficients(the vector representation), and the binary Integer corresponding to the vector representation (the regularrepresentation).

Power	Polynomial	Vector	Regular
0	0	(000)	0
	1	(001)	1
		(010)	2
		(100)	4
		(011)	3
		(110)	6
		(111)	7
		(101)	5

The set of Polynomials in the second column is closed underAddition and Multiplication modulo , and these operations on the set satisfy theAxioms of finite field. This particular finite field is said to be an extension field of degree 3 of GF(2), writtenGF(), and the field GF(2) is called the base field of GF(). If an Irreducible Polynomial generates allelements in this way, it is called a Primitive Irreducible Polynomial. For any Prime or Prime Power and any Positive Integer , there exists a Primitive Irreducible Polynomial ofdegree over GF().

For any element of GF(), , and for any Nonzero element of GF(), . There is a smallestPositive Integer satisfying the sum condition in GF(), which is called the characteristic of thefinite field GF(). The characteristic is a Prime Number for every finite field, and it is true that

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 73-75, 1987.

Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 3rd ed. New York: Macmillan, p. 413, 1965.

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. viii, 1952.

• Deconvolution
• Decreasing Function
• Decreasing Sequence
• Decreasing Series
• Dedekind Cut
• Dedekind Eta Function
• Dedekind Function
• Dedekind Ring
• Dedekind's Axiom
• Dedekind's Problem
• Dedekind Sum
• Deducible
• Deep Theorem
• Defective Matrix
• Defective Number
• Deficiency
• Deficient Number
• Definable Set
• Defined
• Definite Integral
• Degenerate
• Degree
• Degree (Algebraic Surface)
• Degree (Map)
• Degree of Freedom