ExtensionField(F)

ffcat.spad line 34 [edit on github]

• F : Field

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

* : (%, %) -> %

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) -> %

x/y divides x by the scalar y.

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

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.

^ : (%, Integer) -> %

from DivisionRing

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

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 : F -> %

from CoercibleFrom(F)

coerce : Fraction(Integer) -> %

from Algebra(Fraction(Integer))

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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.

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

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%)

from GcdDomain

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

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

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

~= : (%, %) -> Boolean

from BasicType

IntegralDomain

Module(Fraction(Integer))

noZeroDivisors

LeftModule(Fraction(Integer))

RightModule(F)

Algebra(%)

RightModule(%)

Monoid

GcdDomain

AbelianMonoid

UniqueFactorizationDomain

EuclideanDomain

EntireRing

NonAssociativeSemiRng

NonAssociativeAlgebra(Fraction(Integer))

RetractableTo(F)

CancellationAbelianMonoid

MagmaWithUnit

RightModule(Fraction(Integer))

unitsKnown

LeftModule(F)

LeftModule(%)

canonicalUnitNormal

Module(%)

Module(F)

SetCategory

LeftOreRing

CoercibleTo(OutputForm)

Algebra(Fraction(Integer))

Rng

Field

CommutativeRing

TwoSidedRecip

Magma

NonAssociativeRing

SemiGroup

CoercibleFrom(F)

BiModule(F, F)

DivisionRing

BiModule(%, %)

AbelianGroup

AbelianSemiGroup

CommutativeStar

NonAssociativeSemiRing

canonicalsClosed

NonAssociativeAlgebra(%)

Ring

PrincipalIdealDomain

FieldOfPrimeCharacteristic

BiModule(Fraction(Integer), Fraction(Integer))

CharacteristicNonZero

NonAssociativeRng

SemiRng

CharacteristicZero

BasicType

SemiRing