Field

catdef.spad line 565 [edit on github]

The category of commutative fields, i.e. commutative rings where all non-zero elements have multiplicative inverses. The factor operation while trivial is useful to have defined.

* : (%, %) -> %

from Magma

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

from RightModule(Fraction(Integer))

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

from LeftModule(Fraction(Integer))

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

/ : (%, %) -> %

x/y divides the element x by the element y. Error: if y is 0.

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

^ : (%, Integer) -> %

from DivisionRing

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

associator : (%, %, %) -> %

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : % -> %

from Algebra(%)

coerce : Fraction(Integer) -> %

from Algebra(Fraction(Integer))

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

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

factor : % -> Factored(%)

from UniqueFactorizationDomain

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

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

from GcdDomain

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

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

prime? : % -> Boolean

from UniqueFactorizationDomain

principalIdeal : List(%) -> Record(coef : List(%), generator : %)

from PrincipalIdealDomain

quo : (%, %) -> %

from EuclideanDomain

recip : % -> Union(%, "failed")

from MagmaWithUnit

rem : (%, %) -> %

from EuclideanDomain

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

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

Algebra(%)

RightModule(%)

Monoid

LeftModule(Fraction(Integer))

GcdDomain

AbelianMonoid

EuclideanDomain

NonAssociativeAlgebra(Fraction(Integer))

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

AbelianGroup

RightModule(Fraction(Integer))

LeftModule(%)

canonicalUnitNormal

CommutativeStar

Module(%)

SetCategory

LeftOreRing

Algebra(Fraction(Integer))

Rng

CommutativeRing

TwoSidedRecip

Magma

UniqueFactorizationDomain

SemiGroup

BiModule(%, %)

CoercibleTo(OutputForm)

AbelianSemiGroup

NonAssociativeSemiRing

DivisionRing

canonicalsClosed

NonAssociativeAlgebra(%)

PrincipalIdealDomain

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

NonAssociativeRng

unitsKnown

Ring

SemiRng

EntireRing

NonAssociativeSemiRng

BasicType

SemiRing