ExtensionField FΒΆ

ffcat.spad line 34

ExtensionField F is the category of fields which extend the field F

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, F) -> %
from RightModule F
*: (%, Fraction Integer) -> %
from RightModule Fraction Integer
*: (F, %) -> %
from LeftModule F
*: (Fraction Integer, %) -> %
from LeftModule Fraction Integer
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, %) -> %
from Field
/: (%, F) -> %
from VectorSpace F
=: (%, %) -> Boolean
from BasicType
^: (%, Integer) -> %
from DivisionRing
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
algebraic?: % -> Boolean
algebraic?(a) tests whether an element a is algebraic with respect to the ground field F.
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
associates?: (%, %) -> Boolean
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if F has Finite or F has CharacteristicNonZero
from CharacteristicNonZero
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: F -> %
from RetractableTo F
coerce: Fraction Integer -> %
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
commutator: (%, %) -> %
from NonAssociativeRng
degree: % -> OnePointCompletion PositiveInteger
degree(a) returns the degree of minimal polynomial of an element a if a is algebraic with respect to the ground field F, and infinity otherwise.
dimension: () -> CardinalNumber
from VectorSpace F
discreteLog: (%, %) -> Union(NonNegativeInteger, failed) if F has Finite or F has CharacteristicNonZero
from FieldOfPrimeCharacteristic
divide: (%, %) -> Record(quotient: %, remainder: %)
from EuclideanDomain
euclideanSize: % -> NonNegativeInteger
from EuclideanDomain
expressIdealMember: (List %, %) -> Union(List %, failed)
from PrincipalIdealDomain
exquo: (%, %) -> Union(%, failed)
from EntireRing
extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)
from EuclideanDomain
extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)
from EuclideanDomain
extensionDegree: () -> OnePointCompletion PositiveInteger
extensionDegree() returns the degree of the field extension if the extension is algebraic, and infinity if it is not.
factor: % -> Factored %
from UniqueFactorizationDomain
Frobenius: % -> % if F has Finite
Frobenius(a) returns a ^ q where q is the size()\$F.
Frobenius: (%, NonNegativeInteger) -> % if F has Finite
Frobenius(a, s) returns a^(q^s) where q is the size()$F.
gcd: (%, %) -> %
from GcdDomain
gcd: List % -> %
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from GcdDomain
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
inGroundField?: % -> Boolean
inGroundField?(a) tests whether an element a is already in the ground field F.
inv: % -> %
from DivisionRing
latex: % -> String
from SetCategory
lcm: (%, %) -> %
from GcdDomain
lcm: List % -> %
from GcdDomain
lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)
from LeftOreRing
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
multiEuclidean: (List %, %) -> Union(List %, failed)
from EuclideanDomain
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
order: % -> OnePointCompletion PositiveInteger if F has Finite or F has CharacteristicNonZero
from FieldOfPrimeCharacteristic
prime?: % -> Boolean
from UniqueFactorizationDomain
primeFrobenius: % -> % if F has Finite or F has CharacteristicNonZero
from FieldOfPrimeCharacteristic
primeFrobenius: (%, NonNegativeInteger) -> % if F has Finite or F has CharacteristicNonZero
from FieldOfPrimeCharacteristic
principalIdeal: List % -> Record(coef: List %, generator: %)
from PrincipalIdealDomain
quo: (%, %) -> %
from EuclideanDomain
recip: % -> Union(%, failed)
from MagmaWithUnit
rem: (%, %) -> %
from EuclideanDomain
retract: % -> F
from RetractableTo F
retractIfCan: % -> Union(F, failed)
from RetractableTo F
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
sizeLess?: (%, %) -> Boolean
from EuclideanDomain
squareFree: % -> Factored %
from UniqueFactorizationDomain
squareFreePart: % -> %
from UniqueFactorizationDomain
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
transcendenceDegree: () -> NonNegativeInteger
transcendenceDegree() returns the transcendence degree of the field extension, 0 if the extension is algebraic.
transcendent?: % -> Boolean
transcendent?(a) tests whether an element a is transcendent with respect to the ground field F.
unit?: % -> Boolean
from EntireRing
unitCanonical: % -> %
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(F, F)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if F has Finite or F has CharacteristicNonZero

CharacteristicZero if F has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

DivisionRing

EntireRing

EuclideanDomain

Field

FieldOfPrimeCharacteristic if F has Finite or F has CharacteristicNonZero

GcdDomain

IntegralDomain

LeftModule %

LeftModule F

LeftModule Fraction Integer

LeftOreRing

Magma

MagmaWithUnit

Module %

Module F

Module Fraction Integer

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PrincipalIdealDomain

RetractableTo F

RightModule %

RightModule F

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

UniqueFactorizationDomain

unitsKnown

VectorSpace F