# ExtensionField FΒΆ

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

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, F) -> %

from RightModule F

*: (%, Fraction Integer) -> %
*: (F, %) -> %

from LeftModule F

*: (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`.

=: (%, %) -> 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: (%, %) -> %
associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if F has Finite or F has CharacteristicNonZero
coerce: % -> %

from Algebra %

coerce: % -> OutputForm
coerce: F -> %

from CoercibleFrom F

coerce: Fraction Integer -> %
coerce: Integer -> %
commutator: (%, %) -> %
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
divide: (%, %) -> Record(quotient: %, remainder: %)

from EuclideanDomain

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

expressIdealMember: (List %, %) -> Union(List %, failed)
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 %
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

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
plenaryPower: (%, PositiveInteger) -> %
prime?: % -> Boolean
primeFrobenius: % -> % if F has Finite or F has CharacteristicNonZero
primeFrobenius: (%, NonNegativeInteger) -> % if F has Finite or F has CharacteristicNonZero
principalIdeal: List % -> Record(coef: List %, generator: %)
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 %
squareFreePart: % -> %
subtractIfCan: (%, %) -> Union(%, failed)
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

BasicType

BiModule(%, %)

BiModule(F, F)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if F has Finite or F has CharacteristicNonZero

CommutativeRing

CommutativeStar

DivisionRing

EntireRing

EuclideanDomain

Field

FieldOfPrimeCharacteristic if F has Finite or F has CharacteristicNonZero

GcdDomain

IntegralDomain

LeftOreRing

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PrincipalIdealDomain

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown