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- Finite_field abstract "In algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements, called its order (the size of the underlying set). As with any field, a finite field is a set on which the operations of commutative multiplication, addition, subtraction and division (by anything except zero) have been defined. Common, but not the only, examples of finite fields are given by the integers modulo a prime, that is, the integers mod n where n is a prime number, such as ℤ/3ℤ or ℤ/7ℤ.Finite fields only exist when the order (size) is a prime power pk (where p is a prime number and k is a positive integer). For each prime power, there is a finite field with this size, and all fields of a given order are isomorphic. The characteristic of a field of order pk is p (this means that adding p copies of any element always results in zero). ℤ/2ℤ (the integers mod 2) has characteristic 2 since 1 + 1 = 0, while ℤ/5ℤ has characteristic 5 since 0 = 1 + 1 + 1 + 1 + 1 = 2 + 2 + 2 + 2 + 2 = etc. In a finite field of order q, the polynomial Xq − X has all the elements of the finite field as roots, and so, is the product of q different linear factors. Just considering multiplication, the non-zero elements of any finite field form a multiplicative group that is a cyclic group. Therefore, the non-zero elements can be expressed as the powers of a single element called a primitive element of the field (in general there will be several primitive elements for a given field.)A field has, by definition, a commutative multiplication operation. A more general algebraic structure that satisfies all the other axioms of a field but isn't required to have a commutative multiplication is called a division ring (or sometimes skewfield). A finite division ring is a finite field by Wedderburn's little theorem. This result shows that the finiteness condition in the definition of a finite field can have algebraic consequences.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.".
- Finite_field wikiPageExternalLink FiniteField.html.
- Finite_field wikiPageID "11615".
- Finite_field wikiPageRevisionID "606530008".
- Finite_field hasPhotoCollection Finite_field.
- Finite_field subject Category:Field_theory.
- Finite_field subject Category:Finite_fields.
- Finite_field type Field108569998.
- Finite_field type FiniteFields.
- Finite_field type GeographicalArea108574314.
- Finite_field type Location100027167.
- Finite_field type Object100002684.
- Finite_field type PhysicalEntity100001930.
- Finite_field type Region108630985.
- Finite_field type Tract108673395.
- Finite_field type YagoGeoEntity.
- Finite_field type YagoLegalActorGeo.
- Finite_field type YagoPermanentlyLocatedEntity.
- Finite_field comment "In algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements, called its order (the size of the underlying set). As with any field, a finite field is a set on which the operations of commutative multiplication, addition, subtraction and division (by anything except zero) have been defined.".
- Finite_field label "Campo finito".
- Finite_field label "Ciało skończone".
- Finite_field label "Corpo finito".
- Finite_field label "Corps fini".
- Finite_field label "Cuerpo finito".
- Finite_field label "Eindig lichaam (Ned) / Eindig veld (Be)".
- Finite_field label "Endlicher Körper".
- Finite_field label "Finite field".
- Finite_field label "Конечное поле".
- Finite_field label "حقل منته".
- Finite_field label "有限体".
- Finite_field label "有限域".
- Finite_field sameAs Konečné_těleso.
- Finite_field sameAs Endlicher_Körper.
- Finite_field sameAs Πεπερασμένο_σώμα.
- Finite_field sameAs Cuerpo_finito.
- Finite_field sameAs Corps_fini.
- Finite_field sameAs Campo_finito.
- Finite_field sameAs 有限体.
- Finite_field sameAs 유한체.
- Finite_field sameAs _Eindig_veld_(Be).
- Finite_field sameAs Ciało_skończone.
- Finite_field sameAs Corpo_finito.
- Finite_field sameAs m.032gk.
- Finite_field sameAs Q603880.
- Finite_field sameAs Q603880.
- Finite_field sameAs Finite_field.
- Finite_field wasDerivedFrom Finite_field?oldid=606530008.
- Finite_field isPrimaryTopicOf Finite_field.