GF(p)
E641824
GF(p) is a finite field consisting of p elements, where p is a prime number, that forms the basic setting for modular arithmetic and many algebraic and cryptographic constructions.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Gf (math library) | 1 |
| finite field F_7 | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Galois field
ⓘ
commutative ring with identity ⓘ finite field ⓘ integral domain ⓘ |
| additionOperation | addition modulo p ⓘ |
| additiveGroup | cyclic group of order p ⓘ |
| additiveIdentity | 0 ⓘ |
| hasAdditiveOrderOf1 | p ⓘ |
| hasAutomorphismGroup | trivial (only identity) ⓘ |
| hasCardinality | p ⓘ |
| hasCharacteristic | p ⓘ |
| hasElements | equivalence classes of integers modulo p ⓘ |
| hasFrobeniusEndomorphism | x -> x^p ⓘ |
| hasNoZeroDivisors | true ⓘ |
| hasPolynomialRing | GF(p)[x] ⓘ |
| isAlsoKnownAs |
F_p
ⓘ
Z/pZ ⓘ integers modulo p ⓘ |
| isBaseFieldFor | finite field extensions GF(p^n) ⓘ |
| isCommutativeUnderAddition | true ⓘ |
| isCommutativeUnderMultiplication | true ⓘ |
| isConstructedAs | quotient ring Z/pZ ⓘ |
| isDefinedFor | prime number p ⓘ |
| isExampleOf | simple algebraic structure used in cryptographic protocols ⓘ |
| isFinite | true ⓘ |
| isGaloisOver | its prime field ⓘ |
| isInfinite | false ⓘ |
| isIsomorphicTo |
Z/pZ
ⓘ
prime field of characteristic p ⓘ |
| isPerfectField | true ⓘ |
| isPrimeField | true ⓘ |
| isSimpleField | true ⓘ |
| isSmallestFieldOfCharacteristic | p ⓘ |
| isSubfieldOf | any field of characteristic p ⓘ |
| isUsedIn |
algebraic geometry over finite fields
ⓘ
coding theory ⓘ cryptography ⓘ discrete logarithm based cryptosystems ⓘ elliptic curve cryptography ⓘ error-correcting codes ⓘ modular arithmetic ⓘ number theory ⓘ |
| isUsedToDefine | residue classes modulo p ⓘ |
| multiplicationOperation | multiplication modulo p ⓘ |
| multiplicativeGroup | cyclic group of order p-1 ⓘ |
| multiplicativeIdentity | 1 ⓘ |
| satisfiesFieldAxioms | true ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
finite field F_7