GF(p^m)
E643835
GF(p^m) is a finite field with p^m elements, where p is a prime and m is a positive integer, widely used in algebra, coding theory, and cryptography.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| GF(2^8) | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Galois field
ⓘ
finite field ⓘ |
| additiveGroupIs | elementary abelian p-group ⓘ |
| additiveGroupOrder | p^m ⓘ |
| automorphismGroupIs | generated by Frobenius automorphism ⓘ |
| basisType |
dual basis
ⓘ
normal basis ⓘ |
| basisType | polynomial basis ⓘ |
| characteristic | p ⓘ |
| constructedAs | GF(p)[x]/(f(x)) ⓘ |
| constructionCondition | f(x) irreducible of degree m over GF(p) ⓘ |
| definedOver | prime p ⓘ |
| everyElementSatisfies | x^(p^m) = x ⓘ |
| existsIf | p is prime and m is positive integer ⓘ |
| extensionDegree | m ⓘ |
| FrobeniusAutomorphism | x -> x^p ⓘ |
| hasAdditiveIdentity | true ⓘ |
| hasCardinality | p^m ⓘ |
| hasMultiplicativeIdentity | true ⓘ |
| hasSubfield | GF(p^d) for each d dividing m ⓘ |
| isAlgebraicOver | GF(p) ⓘ |
| isCommutative | true ⓘ |
| isExtensionOf | GF(p) ⓘ |
| isFinite | true ⓘ |
| isIntegralDomain | true ⓘ |
| isPerfectField | true ⓘ |
| isSimpleExtension | GF(p)(α) where α is root of irreducible polynomial ⓘ |
| isUniqueUpTo | field isomorphism ⓘ |
| isVectorSpaceOver | GF(p) NERFINISHED ⓘ |
| multiplicativeGroupIs | cyclic group ⓘ |
| multiplicativeGroupOrder | p^m - 1 ⓘ |
| normMapTo | GF(p) ⓘ |
| parameter |
positive integer m
ⓘ
prime p ⓘ |
| polynomialRepresentation | polynomials of degree < m over GF(p) ⓘ |
| subfieldsCorrespondTo | divisors of m ⓘ |
| traceMapTo | GF(p) ⓘ |
| usedIn |
BCH codes
NERFINISHED
ⓘ
Reed–Solomon codes NERFINISHED ⓘ algebra ⓘ block ciphers ⓘ coding theory ⓘ cryptography ⓘ elliptic curve cryptography ⓘ error-correcting codes ⓘ stream ciphers ⓘ |
| vectorSpaceDimension | m ⓘ |
| zeroDivisors | none ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
GF(2^8)