RealClosure(TheField)

reclos.spad line 866 [edit on github]

This domain implements the real closure of an ordered field.

* : (%, %) -> %
from Magma
* : (%, TheField) -> %
from RightModule(TheField)
* : (%, Fraction(Integer)) -> %
from RightModule(Fraction(Integer))
* : (%, Integer) -> %
from RightModule(Integer)
* : (TheField, %) -> %
from LeftModule(TheField)
* : (Fraction(Integer), %) -> %
from LeftModule(Fraction(Integer))
* : (Integer, %) -> %
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
+ : (%, %) -> %
from AbelianSemiGroup
- : % -> %
from AbelianGroup
- : (%, %) -> %
from AbelianGroup
/ : (%, %) -> %
from Field
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
< : (%, %) -> Boolean
from PartialOrder
<= : (%, %) -> Boolean
from PartialOrder
= : (%, %) -> Boolean
from BasicType
> : (%, %) -> Boolean
from PartialOrder
>= : (%, %) -> Boolean
from PartialOrder
^ : (%, Fraction(Integer)) -> %
from RadicalCategory
^ : (%, Integer) -> %
from DivisionRing
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
abs : % -> %
from OrderedRing
algebraicOf : (RightOpenIntervalRootCharacterization(%, SparseUnivariatePolynomial(%)), OutputForm) -> %

algebraicOf(char) is the external number

allRootsOf : Polynomial(%) -> List(%)
from RealClosedField
allRootsOf : Polynomial(Fraction(Integer)) -> List(%)
from RealClosedField
allRootsOf : Polynomial(Integer) -> List(%)
from RealClosedField
allRootsOf : SparseUnivariatePolynomial(%) -> List(%)
from RealClosedField
allRootsOf : SparseUnivariatePolynomial(Fraction(Integer)) -> List(%)
from RealClosedField
allRootsOf : SparseUnivariatePolynomial(Integer) -> List(%)
from RealClosedField
annihilate? : (%, %) -> Boolean
from Rng
antiCommutator : (%, %) -> %
from NonAssociativeSemiRng
approximate : (%, %) -> Fraction(Integer)
from RealClosedField
associates? : (%, %) -> Boolean
from EntireRing
associator : (%, %, %) -> %
from NonAssociativeRng
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
coerce : % -> %
from Algebra(%)
coerce : TheField -> %
from CoercibleFrom(TheField)
coerce : Fraction(Integer) -> %
from CoercibleFrom(Fraction(Integer))
coerce : Integer -> %
from CoercibleFrom(Integer)
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
mainCharacterization : % -> Union(RightOpenIntervalRootCharacterization(%, SparseUnivariatePolynomial(%)), "failed")

mainCharacterization(x) is the main algebraic quantity of x (SEG)

mainDefiningPolynomial : % -> Union(SparseUnivariatePolynomial(%), "failed")
from RealClosedField
mainForm : % -> Union(OutputForm, "failed")
from RealClosedField
mainValue : % -> Union(SparseUnivariatePolynomial(%), "failed")
from RealClosedField
max : (%, %) -> %
from OrderedSet
min : (%, %) -> %
from OrderedSet
multiEuclidean : (List(%), %) -> Union(List(%), "failed")
from EuclideanDomain
negative? : % -> Boolean
from OrderedRing
nthRoot : (%, Integer) -> %
from RadicalCategory
one? : % -> Boolean
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
plenaryPower : (%, PositiveInteger) -> %
from NonAssociativeAlgebra(Integer)
positive? : % -> Boolean
from OrderedRing
prime? : % -> Boolean
from UniqueFactorizationDomain
principalIdeal : List(%) -> Record(coef : List(%), generator : %)
from PrincipalIdealDomain
quo : (%, %) -> %
from EuclideanDomain
recip : % -> Union(%, "failed")
from MagmaWithUnit
relativeApprox : (%, %) -> Fraction(Integer)

relativeApprox(n, p) gives a relative approximation of n that has precision p

rem : (%, %) -> %
from EuclideanDomain
rename : (%, OutputForm) -> %
from RealClosedField
rename! : (%, OutputForm) -> %
from RealClosedField
retract : % -> TheField
from RetractableTo(TheField)
retract : % -> Fraction(Integer)
from RetractableTo(Fraction(Integer))
retract : % -> Integer
from RetractableTo(Integer)
retractIfCan : % -> Union(TheField, "failed")
from RetractableTo(TheField)
retractIfCan : % -> Union(Fraction(Integer), "failed")
from RetractableTo(Fraction(Integer))
retractIfCan : % -> Union(Integer, "failed")
from RetractableTo(Integer)
rightPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower : (%, PositiveInteger) -> %
from Magma
rightRecip : % -> Union(%, "failed")
from MagmaWithUnit
rootOf : (SparseUnivariatePolynomial(%), PositiveInteger) -> Union(%, "failed")
from RealClosedField
rootOf : (SparseUnivariatePolynomial(%), PositiveInteger, OutputForm) -> Union(%, "failed")
from RealClosedField
sample : () -> %
from AbelianMonoid
sign : % -> Integer
from OrderedRing
sizeLess? : (%, %) -> Boolean
from EuclideanDomain
smaller? : (%, %) -> Boolean
from Comparable
sqrt : % -> %
from RealClosedField
sqrt : (%, PositiveInteger) -> %
from RealClosedField
sqrt : Fraction(Integer) -> %
from RealClosedField
sqrt : Integer -> %
from RealClosedField
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

Module(Fraction(Integer))

PrincipalIdealDomain

PartialOrder

NonAssociativeSemiRing

BiModule(%, %)

Field

canonicalUnitNormal

Rng

RealClosedField

CoercibleFrom(Integer)

TwoSidedRecip

SemiRing

EntireRing

LeftOreRing

NonAssociativeAlgebra(Fraction(Integer))

unitsKnown

RadicalCategory

BiModule(TheField, TheField)

Algebra(Integer)

LeftModule(Integer)

noZeroDivisors

RetractableTo(Fraction(Integer))

OrderedSet

UniqueFactorizationDomain

SemiGroup

NonAssociativeAlgebra(Integer)

RightModule(Fraction(Integer))

Magma

GcdDomain

IntegralDomain

LeftModule(%)

CharacteristicZero

Algebra(%)

FullyRetractableTo(TheField)

CommutativeRing

NonAssociativeAlgebra(TheField)

OrderedAbelianMonoid

DivisionRing

NonAssociativeSemiRng

CancellationAbelianMonoid

EuclideanDomain

canonicalsClosed

Comparable

RetractableTo(Integer)

OrderedCancellationAbelianMonoid

OrderedRing

RightModule(TheField)

CommutativeStar

AbelianMonoid

RetractableTo(TheField)

MagmaWithUnit

NonAssociativeRing

RightModule(%)

OrderedAbelianSemiGroup

Module(%)

CoercibleTo(OutputForm)

LeftModule(TheField)

BiModule(Integer, Integer)

Module(Integer)

SemiRng

Monoid

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

BasicType

Ring

RightModule(Integer)

LeftModule(Fraction(Integer))

AbelianSemiGroup

SetCategory

FullyRetractableTo(Fraction(Integer))

Algebra(TheField)

CoercibleFrom(Fraction(Integer))

NonAssociativeRng

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

OrderedAbelianGroup

Module(TheField)

AbelianGroup

CoercibleFrom(TheField)