EuclideanDomain¶

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A constructive euclidean domain, i.e. one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (euclideanSize) than the divisor. Conditional attributes: multiplicativeValuation Size(a*b)=Size(a)*Size(b) additiveValuation Size(a*b)=Size(a)+Size(b)

0: %: from AbelianMonoid

1: %: from MagmaWithUnit

*: (%, %) -> %: from Magma
*: (Integer, %) -> %: from AbelianGroup
*: (NonNegativeInteger, %) -> %: from AbelianMonoid
*: (PositiveInteger, %) -> %: from AbelianSemiGroup

+: (%, %) -> %: from AbelianSemiGroup

-: % -> %: from AbelianGroup
-: (%, %) -> %: from AbelianGroup

=: (%, %) -> Boolean: from BasicType

^: (%, NonNegativeInteger) -> %: from MagmaWithUnit
^: (%, PositiveInteger) -> %: from Magma

~=: (%, %) -> Boolean: from BasicType

annihilate?: (%, %) -> Boolean: from Rng

antiCommutator: (%, %) -> %: from NonAssociativeSemiRng

associates?: (%, %) -> Boolean: from EntireRing

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

characteristic: () -> NonNegativeInteger: from NonAssociativeRing

coerce: % -> %: from Algebra %
coerce: % -> OutputForm: from CoercibleTo OutputForm
coerce: Integer -> %: from NonAssociativeRing

commutator: (%, %) -> %: from NonAssociativeRng

divide: (%, %) -> Record(quotient: %, remainder: %): divide(x, y) divides x by y producing a record containing a quotient and remainder, where the remainder is smaller (see sizeLess?) than the divisor y.

euclideanSize: % -> NonNegativeInteger: euclideanSize(x) returns the euclidean size of the element x. Error: if x is zero.

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

exquo: (%, %) -> Union(%, failed): from EntireRing

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %): extendedEuclidean(x, y) returns a record rec where rec.coef1*x+rec.coef2*y = rec.generator and rec.generator is a gcd of x and y. The gcd is unique only up to associates if canonicalUnitNormal is not asserted. principalIdeal provides a version of this operation which accepts an arbitrary length list of arguments.

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed): extendedEuclidean(x, y, z) either returns a record rec where rec.coef1*x+rec.coef2*y=z or returns “failed” if z cannot be expressed as a linear combination of x and y.

gcd: (%, %) -> %: from GcdDomain
gcd: List % -> %: from GcdDomain

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

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): multiEuclidean([f1, ..., fn], z) returns a list of coefficients [a1, ..., an] such that `` z / prod fi = sum aj/fj``. If no such list of coefficients exists, “failed” is returned.

one?: % -> Boolean: from MagmaWithUnit

opposite?: (%, %) -> Boolean: from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %: from NonAssociativeAlgebra %

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

quo: (%, %) -> %: x quo y is the same as divide(x, y).quotient. See divide.

recip: % -> Union(%, failed): from MagmaWithUnit

rem: (%, %) -> %: x rem y is the same as divide(x, y).remainder. See divide.

rightPower: (%, NonNegativeInteger) -> %: from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %: from Magma

rightRecip: % -> Union(%, failed): from MagmaWithUnit

sample: %: from AbelianMonoid

sizeLess?: (%, %) -> Boolean: sizeLess?(x, y) tests whether x is strictly smaller than y with respect to the euclideanSize. Note: zero is considered smaller than every nonzero element.