# Number Fields and Orders - [Introduction](introduction.md) - [Types](introduction.md#types) - [Acknowledgement](acknowledgement.md) - [Creation Functions](creation.md) - [Creation of Algebraic Fields](creation.md#creation-of-algebraic-fields) - [`NumberField(f): RngUPolElt → FldNum`](creation.md#function-numberfield-rngupolelt) - [`RationalsAsNumberField() → FldNum`](creation.md#function-rationalsasnumberfield) - [`QNF() → FldNum`](creation.md#function-qnf) - [`NumberField(s): [ RngUPolElt ] → FldNum`](creation.md#function-numberfield-rngupolelt-2) - [`ext< F | s1, ..., sn >: FldAlg, RngUPolElt, ..., RngUPolElt → FldAlg`](creation.md#constructor-constructor-ext-fldalg-rngupolelt-rngupolelt-fldalg) - [`ext: FldAlg, [RngUPolElt] → FldAlg`](creation.md#constructor-constructor-ext-fldalg-rngupolelt-fldalg) - [`ext< Q | s1, ..., sn >: FldRat, RngUPolElt, ..., RngUPolElt → FldNum`](creation.md#constructor-constructor-ext-fldrat-rngupolelt-rngupolelt-fldnum) - [`ext< Q | s >: FldRat, [RngUPolElt] → FldNum`](creation.md#constructor-constructor-ext-fldrat-rngupolelt-fldnum) - [`RadicalExtension(F, d, a): Rng, RngIntElt, RngElt → FldAlg`](creation.md#function-radicalextension-rng-rngintelt-rngelt) - [`SplittingField(F): FldAlg → FldAlg, SeqEnum`](creation.md#function-splittingfield-fldalg) - [`NormalClosure(F): FldAlg → FldAlg, SeqEnum`](creation.md#function-normalclosure-fldalg) - [`SplittingField(f): RngUPolElt → FldAlg`](creation.md#function-splittingfield-rngupolelt) - [`SplittingField(L): [RngUPolElt] → FldNum, [FldNumElt]`](creation.md#function-splittingfield-rngupolelt-2) - [`sub< F | e₁, ..., eₙ >: FldAlg, FldAlgElt, ..., FldAlgElt → FldAlg, Map`](creation.md#constructor-constructor-sub-fldalg-fldalgelt-fldalgelt-fldalg-map) - [`sub< F | S >: FldAlg, SeqEnum → FldAlg, Map`](creation.md#constructor-constructor-sub-fldalg-seqenum-fldalg-map) - [`MergeFields(F, L): FldAlg, FldAlg → SeqEnum`](creation.md#function-mergefields-fldalg-fldalg) - [`CompositeFields(F, L): FldAlg, FldAlg → SeqEnum`](creation.md#function-rngord-compositefields) - [`Compositum(K, L): FldAlg, FldAlg → FldAlg`](creation.md#function-compositum-fldalg-fldalg) - [`Compositum(K, A): FldAlg, FldAb → FldAlg`](creation.md#function-compositum-fldalg-fldab) - [`OptimizedRepresentation(F): FldAlg → FldAlg, Map`](creation.md#function-rngord-optimizedrepresentation) - [`OptimisedRepresentation(F): FldAlg → FldAlg, Map`](creation.md#function-optimisedrepresentation-fldalg) - [`OptimizedRepresentation(F, d): FldAlg, RngIntElt → FldAlg, Map`](creation.md#function-optimizedrepresentation-fldalg-rngintelt) - [`OptimisedRepresentation(F, d): FldAlg, RngIntElt → FldAlg, Map`](creation.md#function-optimisedrepresentation-fldalg-rngintelt) - [`Example: Opt Rep Ord`](creation.md#example-ex-509ba1) - [Creation of Orders and Fields from Orders](creation.md#creation-of-orders-and-fields-from-orders) - [`EquationOrder(f): RngUPolElt → RngOrd`](creation.md#function-equationorder-rngupolelt) - [`EquationOrder(S): [RngUPolElt] → RngOrd`](creation.md#function-equationorder-rngupolelt-2) - [`EquationOrder(K): FldNum → RngOrd`](creation.md#function-equationorder-fldnum) - [`SubOrder(O): RngOrd → RngOrd`](creation.md#function-suborder-rngord) - [`EquationOrder(O): RngOrd → RngOrd`](creation.md#function-equationorder-rngord) - [`Integers(O): RngOrd → RngOrd`](creation.md#function-integers-rngord) - [`RingOfIntegers(O): RngOrd → RngOrd`](creation.md#function-ringofintegers-rngord) - [`IntegerRing(O): RngOrd → RngOrd`](creation.md#function-integerring-rngord) - [`Example: Orders`](creation.md#example-ex-4c8125) - [`sub< O | a₁, ..., aᵣ >: RngOrd, RngOrdElt, ..., RngOrdElt → RngOrd`](creation.md#constructor-constructor-sub-rngord-rngordelt-rngordelt-rngord) - [`ext< O | a₁, ..., aᵣ >: RngOrd, RngOrdElt, ..., RngOrdElt → RngOrd`](creation.md#constructor-rngord-ext-seq) - [`ext< Z | f >: RngInt, RngUPolElt → RngOrd`](creation.md#constructor-constructor-ext-rngint-rngupolelt-rngord) - [`ext< O | f >: RngOrd, RngUPolElt → RngOrd`](creation.md#constructor-constructor-ext-rngord-rngupolelt-rngord) - [`FieldOfFractions(O): RngOrd → FldOrd`](creation.md#function-rngord-fieldoffractions) - [`Order(F): FldOrd → RngOrd`](creation.md#function-order-fldord) - [`NumberField(O): RngOrd → FldNum`](creation.md#function-numberfield-rngord) - [`NumberField(F): FldOrd → FldNum`](creation.md#function-numberfield-fldord) - [`Example: fractions`](creation.md#example-ex-94a3f9) - [`OptimizedRepresentation(O): RngOrd → BoolElt, RngOrd, Map`](creation.md#function-optimizedrepresentation-rngord) - [`OptimisedRepresentation(O): RngOrd → BoolElt, RngOrd, Map`](creation.md#function-optimisedrepresentation-rngord) - [`OptimizedRepresentation(O, d): RngOrd, RngIntElt → BoolElt, RngOrd, Map`](creation.md#function-optimizedrepresentation-rngord-rngintelt) - [`OptimisedRepresentation(O, d): RngOrd, RngIntElt → BoolElt, RngOrd, Map`](creation.md#function-optimisedrepresentation-rngord-rngintelt) - [`O + P: RngOrd, RngOrd → RngOrd`](creation.md#operation-op-plus-rngord-rngord) - [`O meet P: RngOrd, RngOrd → RngOrd`](creation.md#operation-op-meet-rngord-rngord) - [`AsExtensionOf(O, P): RngOrd, RngOrd → RngOrd`](creation.md#function-asextensionof-rngord-rngord) - [`Order(O, T, d): RngOrd, AlgMatElt, RngIntElt → RngOrd`](creation.md#function-order-rngord-algmatelt-rngintelt) - [`Order(O, M): RngOrd, ModDed → RngOrd`](creation.md#function-order-rngord-modded) - [`Order(O, M): RngOrd, PMat → RngOrd`](creation.md#function-order-rngord-pmat) - [`Order( [ e₁, ... eₙ ] ): [FldAlgElt] → RngOrd`](creation.md#function-rngord-order-seq) - [Maximal Orders](creation.md#maximal-orders) - [`MaximalOrder(O): RngOrd → RngOrd`](creation.md#function-maximalorder-rngord) - [`MaximalOrder(F): FldAlg → RngOrd`](creation.md#function-rngord-maximalorder) - [`IntegerRing(F): FldAlg → RngOrd`](creation.md#function-integerring-fldalg) - [`Integers(F): FldAlg → RngOrd`](creation.md#function-integers-fldalg) - [`RingOfIntegers(F): FldAlg → RngOrd`](creation.md#function-ringofintegers-fldalg) - [`MaximalOrder(f): RngUPolElt → RngOrd`](creation.md#function-rngord-maxord) - [`Example: Max Order`](creation.md#example-ex-474b81) - [Orders and Ideals](creation.md#orders-and-ideals) - [`pMaximalOrder(O, p): RngOrd, RngIntElt → RngOrd`](creation.md#function-pmaximalorder-rngord-rngintelt) - [`pMaximalOrder(O, p): RngOrd, RngOrdIdl → RngOrd`](creation.md#function-pmaximalorder-rngord-rngordidl) - [`pRadical(O, p): RngOrd, RngIntElt → RngOrdIdl`](creation.md#function-pradical-rngord-rngintelt) - [`pRadical(O, p): RngOrd, RngOrdIdl → RngOrdIdl`](creation.md#function-pradical-rngord-rngordidl) - [`MultiplicatorRing(I): RngOrdFracIdl → Rng`](creation.md#function-multiplicatorring-rngordfracidl) - [`Example: Round2`](creation.md#example-ex-638433) - [Creation of Elements](creation.md#creation-of-elements) - [`F ! a: FldAlg, RngElt → FldAlgElt`](creation.md#operation-op-fldalg-rngelt) - [`elt< F | a >: FldAlg, RngElt → FldAlgElt`](creation.md#constructor-constructor-elt-fldalg-rngelt-fldalgelt) - [`F ! [a₀, a₁, ..., aₘ₋₁]: FldAlg, [RngElt] → FldAlgElt`](creation.md#operation-op-fldalg-rngelt-2) - [`elt< F | [ a₀, a₁, ..., aₘ₋₁ ] >: FldAlg, SeqEnum[RngElt] → FldAlgElt`](creation.md#constructor-constructor-elt-fldalg-seqenum-rngelt-fldalgelt) - [`elt< F | a₀, a₁, ..., aₘ₋₁>: FldAlg, RngElt, ..., RngElt → FldAlgElt`](creation.md#constructor-constructor-elt-fldalg-rngelt-rngelt-fldalgelt) - [`O ! a: RngOrd, RngElt → RngOrdElt`](creation.md#operation-op-rngord-rngelt) - [`elt< O | a >: RngOrd, RngElt → RngOrdElt`](creation.md#constructor-constructor-elt-rngord-rngelt-rngordelt) - [`O ! [a₀, a₁, ..., aₘ₋₁]: RngOrd, [ RngElt ] → RngOrdElt`](creation.md#operation-op-rngord-rngelt-2) - [`elt< O | [ a₀, a₁, ..., aₘ₋₁ ] >: RngOrd, SeqEnum[RngElt] → RngOrdElt`](creation.md#constructor-constructor-elt-rngord-seqenum-rngelt-rngordelt) - [`elt< O | a₀, a₁, ..., aₘ₋₁ >: RngOrd, RngElt, ..., RngElt → RngOrdElt`](creation.md#constructor-constructor-elt-rngord-rngelt-rngelt-rngordelt) - [`Random(F, m): FldAlg, RngIntElt → FldAlgElt`](creation.md#function-random-fldalg-rngintelt) - [`Random(O, m): RngOrd, RngIntElt → RngOrdElt`](creation.md#function-random-rngord-rngintelt) - [`Random(I, m): RngOrdFracIdl, RngIntElt → FldOrdElt`](creation.md#function-random-rngordfracidl-rngintelt) - [`Example: Elements`](creation.md#example-ex-759e86) - [`One(K): FldAlg → FldAlgElt`](creation.md#function-one-fldalg) - [`One(O): RngOrd → RngOrdElt`](creation.md#function-one-rngord) - [`Identity(K): FldAlg → FldAlgElt`](creation.md#function-identity-fldalg) - [`Identity(O): RngOrd → RngOrdElt`](creation.md#function-identity-rngord) - [`Zero(K): FldAlg → FldAlgElt`](creation.md#function-zero-fldalg) - [`Zero(O): RngOrd → RngOrdElt`](creation.md#function-zero-rngord) - [`Representative(K): FldAlg → FldAlgElt`](creation.md#function-representative-fldalg) - [`Representative(O): RngOrd → RngOrdElt`](creation.md#function-representative-rngord) - [Creation of Homomorphisms](creation.md#creation-of-homomorphisms) - [`hom< F -> R | r >: FldAlg, Rng, RngElt → Map`](creation.md#constructor-constructor-hom-fldalg-rng-rngelt-map) - [`hom< F -> R | h, r >: FldAlg, Rng, Map, RngElt → Map`](creation.md#constructor-constructor-hom-fldalg-rng-map-rngelt-map) - [`hom< O -> R | r >: RngOrd, Rng, RngElt → Map`](creation.md#constructor-constructor-hom-rngord-rng-rngelt-map) - [`hom< O -> R | h, r >: RngOrd, Rng, Map, RngElt → Map`](creation.md#constructor-constructor-hom-rngord-rng-map-rngelt-map) - [`Example: Homomorphisms`](creation.md#example-ex-d59686) - [`hom< O -> R | b₁, ..., bₙ >: RngFunOrd, Rng, RngElt, ..., RngElt → Map`](creation.md#constructor-constructor-hom-rngfunord-rng-rngelt-rngelt-map) - [`hom< O -> R | m, b₁, ..., bₙ >: RngFunOrd, Rng, Map, RngElt, ..., RngElt → Map`](creation.md#constructor-constructor-hom-rngfunord-rng-map-rngelt-rngelt-map) - [`IsRingHomomorphism(m): Map → BoolElt`](creation.md#function-isringhomomorphism-map) - [Printing](printing.md) - [`SetVerbose(s, n): MonStgElt, RngIntElt`](printing.md#function-setverbose-monstgelt-rngintelt) - [`SetKantPrinting(f): BoolElt → BoolElt`](printing.md#function-setkantprinting-boolelt) - [Real Precision](precision.md) - [`SetKantPrecision(F, n): FldAlg, RngIntElt`](precision.md#function-setkantprecision-fldalg-rngintelt) - [`SetKantPrecision(F, n, m): FldAlg, RngIntElt, RngIntElt`](precision.md#function-setkantprecision-fldalg-rngintelt-rngintelt) - [`SetKantPrecision(O, n): RngOrd, RngIntElt`](precision.md#function-setkantprecision-rngord-rngintelt) - [`SetKantPrecision(O, n, m): RngOrd, RngIntElt, RngIntElt`](precision.md#function-setkantprecision-rngord-rngintelt-rngintelt) - [Structure Operations](operation.md) - [General Functions](operation.md#general-functions) - [`Category(F): FldAlg → Cat`](operation.md#function-category-fldalg) - [`Parent(F): FldAlg → Pow`](operation.md#function-parent-fldalg) - [`Category(O): RngOrd → Cat`](operation.md#function-category-rngord) - [`Parent(O): RngOrd → Pow`](operation.md#function-parent-rngord) - [`AssignNames(~K, s): FldNum, [ MonStgElt ]`](operation.md#function-assignnames-fldnum-monstgelt-ref) - [`Name(K, i): FldNum, RngIntElt → FldNumElt`](operation.md#function-name-fldnum-rngintelt) - [`K . i: FldNum, RngIntElt → FldNumElt`](operation.md#operation-operation-fldnum-rngintelt-fldnumelt) - [`AssignNames(~F, s): FldOrd, [ MonStgElt ]`](operation.md#function-assignnames-fldord-monstgelt-ref) - [`F . i: FldOrd, RngIntElt → FldOrdElt`](operation.md#operation-operation-fldord-rngintelt-fldordelt) - [`Name(F, i): FldOrd, RngIntElt → FldOrdElt`](operation.md#function-name-fldord-rngintelt) - [`O . i: RngOrd, RngIntElt → FldOrdElt`](operation.md#operation-operation-rngord-rngintelt-fldordelt) - [Related Structures](operation.md#related-structures) - [`GroundField(F): FldAlg → Fld`](operation.md#function-groundfield-fldalg) - [`BaseField(F): FldAlg → Fld`](operation.md#function-basefield-fldalg) - [`CoefficientField(F): FldAlg → Fld`](operation.md#function-coefficientfield-fldalg) - [`CoefficientRing(F): FldAlg → Fld`](operation.md#function-coefficientring-fldalg) - [`BaseRing(O): RngOrd → Rng`](operation.md#function-rngord-basering) - [`CoefficientRing(O): RngOrd → Rng`](operation.md#function-coefficientring-rngord) - [`AbsoluteField(F): FldAlg → FldAlg`](operation.md#function-absolutefield-fldalg) - [`AbsoluteOrder(O): RngOrd → RngOrd`](operation.md#function-absoluteorder-rngord) - [`SimpleExtension(F): FldAlg → FldAlg`](operation.md#function-simpleextension-fldalg) - [`SimpleExtension(O): RngOrd → RngOrd`](operation.md#function-simpleextension-rngord) - [`RelativeField(F, L): FldAlg, FldAlg → FldAlg`](operation.md#function-relativefield-fldalg-fldalg) - [`RelativeField(Q, L): FldRat, FldAlg → FldAlg`](operation.md#function-relativefield-fldrat-fldalg) - [`Components(F): FldAlg → [FldAlg]`](operation.md#function-components-fldalg) - [`Components(O): RngOrd → [RngOrd]`](operation.md#function-components-rngord) - [`Example: Compositum`](operation.md#example-ex-3aa334) - [`Simplify(O): RngOrd → RngOrd`](operation.md#function-simplify-rngord) - [`LLL(O): RngOrd → RngOrd, AlgMatElt`](operation.md#function-lll-rngord) - [`Example: lll`](operation.md#example-ex-06710d) - [`PrimeRing(F): FldAlg → RngRat`](operation.md#function-primering-fldalg) - [`PrimeField(F): FldAlg → RngRat`](operation.md#function-primefield-fldalg) - [`PrimeRing(O): RngOrd → RngInt`](operation.md#function-primering-rngord) - [`Centre(F): FldAlg → FldAlg`](operation.md#function-centre-fldalg) - [`Centre(O): RngOrd → RngOrd`](operation.md#function-centre-rngord) - [`Embed(F, L, a): FldAlg, FldAlg, FldAlgElt`](operation.md#function-embed-fldalg-fldalg-fldalgelt) - [`Embed(F, L, a): FldAlg, FldAlg, [FldAlgElt]`](operation.md#function-embed-fldalg-fldalg-fldalgelt-2) - [`EmbeddingMap(F, L): FldAlg, FldAlg → Map`](operation.md#function-embeddingmap-fldalg-fldalg) - [`HasEmbedding(F, L): FldAlg, FldAlg → Bool`](operation.md#function-hasembedding-fldalg-fldalg) - [`HasEmbedding(O1, O2): RngOrd, RngOrd → Bool`](operation.md#function-hasembedding-rngord-rngord) - [`CanComputeEmbedding(F, L): FldAlg, FldAlg → Bool`](operation.md#function-cancomputeembedding-fldalg-fldalg) - [`CanComputeEmbedding(O1, O2): RngOrd, RngOrd → Bool`](operation.md#function-cancomputeembedding-rngord-rngord) - [`Example: em`](operation.md#example-ex-a44554) - [`Completion(K, P): FldAlg, RngOrdIdl → FldLoc, Map`](operation.md#function-rngord-completion1) - [`Completion(O, P): RngOrd, RngOrdIdl → RngLoc, Map`](operation.md#function-completion-rngord-rngordidl) - [`comp: FldAlg, RngOrdIdl → FldLoc, Map`](operation.md#constructor-constructor-comp-fldalg-rngordidl-fldloc-map) - [`comp: RngOrd, RngOrdIdl → RngLoc, Map`](operation.md#constructor-constructor-comp-rngord-rngordidl-rngloc-map) - [`Completion(K, P): FldAlg, PlcNumElt → FldLoc, Map`](operation.md#function-completion-fldalg-plcnumelt) - [`LocalRing(P, prec): RngOrdIdl, RngIntElt → RngLoc, Map`](operation.md#function-localring-rngordidl-rngintelt) - [`Localization(O, P): RngOrd, RngOrdIdl → RngVal, Map`](operation.md#function-localization-rngord-rngordidl) - [Representing Fields as Vector Spaces](operation.md#representing-fields-as-vector-spaces) - [`Algebra(K, J): FldAlg, Fld → AlgAss, Map`](operation.md#function-algebra-fldalg-fld) - [`Algebra(K, J, S): FldAlg, Fld, [FldAlgElt] → AlgAss, Map`](operation.md#function-algebra-fldalg-fld-fldalgelt) - [`VectorSpace(K, J): FldAlg, Fld → ModTupFld, Map`](operation.md#function-vectorspace-fldalg-fld) - [`KSpace(K, J): FldAlg, Fld → ModTupFld, Map`](operation.md#function-kspace-fldalg-fld) - [`VectorSpace(K, J, S): FldAlg, Fld, [FldAlgElt] → ModTupFld, Map`](operation.md#function-vectorspace-fldalg-fld-fldalgelt) - [`KSpace(K, J, S): FldAlg, Fld, [FldAlgElt] → ModTupFld, Map`](operation.md#function-kspace-fldalg-fld-fldalgelt) - [`Example: Vector Space Eg`](operation.md#example-ex-4636b1) - [Invariants](operation.md#invariants) - [`Characteristic(F): FldAlg → RngIntElt`](operation.md#function-characteristic-fldalg) - [`Characteristic(O): RngOrd → RngIntElt`](operation.md#function-characteristic-rngord) - [`Degree(O): RngOrd → RngIntElt`](operation.md#function-degree-rngord) - [`Degree(F): FldAlg → RngIntElt`](operation.md#function-rngord-degree) - [`AbsoluteDegree(O): RngOrd → RngIntElt`](operation.md#function-absolutedegree-rngord) - [`AbsoluteDegree(F): FldAlg → RngIntElt`](operation.md#function-absolutedegree-fldalg) - [`Discriminant(O): RngOrd → RngIntElt`](operation.md#function-discriminant-rngord) - [`Discriminant(O): RngInt → RngIntElt`](operation.md#function-discriminant-rngint) - [`Discriminant(F): FldAlg → RngIntElt`](operation.md#function-rngord-discriminant) - [`Discriminant(O): RngOrd → RngOrdIdl`](operation.md#function-discriminant-rngord-2) - [`Discriminant(F): FldAlg → RngOrdIdl`](operation.md#function-discriminant-fldalg) - [`AbsoluteDiscriminant(O): RngOrd → RngIntElt`](operation.md#function-absolutediscriminant-rngord) - [`AbsoluteDiscriminant(K): FldAlg → FldRatElt`](operation.md#function-absolutediscriminant-fldalg) - [`ReducedDiscriminant(O): RngOrd → RngIntElt`](operation.md#function-reduceddiscriminant-rngord) - [`ReducedDiscriminant(F): FldAlg → RngIntElt`](operation.md#function-reduceddiscriminant-fldalg) - [`ReducedDiscriminant(O): RngOrd → RngOrdIdl`](operation.md#function-reduceddiscriminant-rngord-2) - [`ReducedDiscriminant(F): FldAlg → RngOrdIdl`](operation.md#function-reduceddiscriminant-fldalg-2) - [`Regulator(O: parameters): RngOrd → FldReElt`](operation.md#function-regulator-rngord) - [`Regulator(K): FldNum → FldReElt`](operation.md#function-regulator-fldnum) - [`RegulatorLowerBound(O): RngOrd → FldReElt`](operation.md#function-regulatorlowerbound-rngord) - [`RegulatorLowerBound(K): FldNum → FldReElt`](operation.md#function-regulatorlowerbound-fldnum) - [`Signature(O): RngOrd → RngIntElt, RngIntElt`](operation.md#function-signature-rngord) - [`Signature(F): FldAlg → RngIntElt, RngIntElt`](operation.md#function-signature-fldalg) - [`UnitRank(O): RngOrd → RngIntElt`](operation.md#function-unitrank-rngord) - [`UnitRank(K): FldNum → RngIntElt`](operation.md#function-unitrank-fldnum) - [`Index(O, S): RngOrd, RngOrd → RngIntElt`](operation.md#function-index-rngord-rngord) - [`Index(O, S): RngOrd, RngOrd → RngOrdIdl`](operation.md#function-index-rngord-rngord-2) - [`DefiningPolynomial(F): FldAlg → RngUPolElt`](operation.md#function-definingpolynomial-fldalg) - [`DefiningPolynomial(O): RngOrd → RngUPolElt`](operation.md#function-definingpolynomial-rngord) - [`DefiningPolynomial(F): FldAlg → [RngUPolElt]`](operation.md#function-definingpolynomial-fldalg-2) - [`DefiningPolynomial(O): RngOrd → [RngUPolElt]`](operation.md#function-definingpolynomial-rngord-2) - [`Zeroes(O, n): RngOrd, RngIntElt → [ FldReElt ]`](operation.md#function-zeroes-rngord-rngintelt) - [`Zeros(O, n): RngOrd, RngIntElt → [ FldReElt ]`](operation.md#function-zeros-rngord-rngintelt) - [`Zeroes(F, n): FldAlg, RngIntElt → [ FldReElt ]`](operation.md#function-zeroes-fldalg-rngintelt) - [`Zeros(F, n): FldAlg, RngIntElt → [ FldReElt ]`](operation.md#function-zeros-fldalg-rngintelt) - [`Example: zero`](operation.md#example-ex-afbcf8) - [`Different(O): RngOrd → RngOrdIdl`](operation.md#function-different-rngord) - [`Conductor(O): RngOrd → RngOrdIdl`](operation.md#function-conductor-rngord) - [Basis Representation](operation.md#basis-representation) - [`Basis(O): RngOrd → [ FldOrdElt ]`](operation.md#function-basis-rngord) - [`Basis(O, R): RngOrd, Rng → [ RngElt ]`](operation.md#function-basis-rngord-rng) - [`Basis(F): FldAlg → [ FldAlgElt ]`](operation.md#function-basis-fldalg) - [`Basis(F, R): FldAlg, Rng → [ RngElt ]`](operation.md#function-basis-fldalg-rng) - [`IntegralBasis(F): FldAlg → [ FldAlgElt ]`](operation.md#function-integralbasis-fldalg) - [`IntegralBasis(F, R): FldAlg, Rng → [ RngElt ]`](operation.md#function-integralbasis-fldalg-rng) - [`Example: Basis Ring`](operation.md#example-ex-6e96c8) - [`AbsoluteBasis(K): FldAlg → [FldAlgElt]`](operation.md#function-rngord-absolutebasis) - [`BasisMatrix(O): RngOrd → AlgMatElt`](operation.md#function-basismatrix-rngord) - [`TransformationMatrix(O, P): RngOrd, RngOrd → AlgMatElt, RngIntElt`](operation.md#function-transformationmatrix-rngord-rngord) - [`CoefficientIdeals(O): RngOrd → [RngOrdFracIdl]`](operation.md#function-coefficientideals-rngord) - [`Example: Bases`](operation.md#example-ex-292651) - [`MultiplicationTable(O): RngOrd → [AlgMatElt]`](operation.md#function-multiplicationtable-rngord) - [`TraceMatrix(O): RngOrd → AlgMatElt`](operation.md#function-tracematrix-rngord) - [`TraceMatrix(F): FldAlg → AlgMatElt`](operation.md#function-tracematrix-fldalg) - [`Example: Multiplication Table`](operation.md#example-ex-8f1664) - [Ring Predicates](operation.md#ring-predicates) - [`N eq O: RngOrd, RngOrd → BoolElt`](operation.md#operation-op-eq-rngord-rngord) - [`F eq L: FldAlg, FldAlg → BoolElt`](operation.md#operation-op-eq-fldalg-fldalg) - [`F eq L: FldRat, FldAlg → BoolElt`](operation.md#operation-op-eq-fldrat-fldalg) - [`F eq L: FldAlg, FldRat → BoolElt`](operation.md#operation-op-eq-fldalg-fldrat) - [`IsCommutative(R): Rng → BoolElt`](operation.md#function-iscommutative-rng) - [`IsUnitary(R): Rng → BoolElt`](operation.md#function-isunitary-rng) - [`IsFinite(R): Rng → BoolElt`](operation.md#function-isfinite-rng) - [`IsOrdered(R): Rng → BoolElt`](operation.md#function-isordered-rng) - [`IsField(R): Rng → BoolElt`](operation.md#function-isfield-rng) - [`IsNumberField(R): . → BoolElt`](operation.md#function-isnumberfield) - [`IsAlgebraicField(R): Any → BoolElt`](operation.md#function-isalgebraicfield-any) - [`IsEuclideanDomain(F): FldAlg → BoolElt`](operation.md#function-iseuclideandomain-fldalg) - [`IsSimple(F): FldAlg → BoolElt`](operation.md#function-issimple-fldalg) - [`IsSimple(O): RngOrd → BoolElt`](operation.md#function-issimple-rngord) - [`IsPID(F): FldAlg → BoolElt`](operation.md#function-ispid-fldalg) - [`IsUFD(F): FldAlg → BoolElt`](operation.md#function-isufd-fldalg) - [`IsPrincipalIdealRing(F): FldAlg → BoolElt`](operation.md#function-isprincipalidealring-fldalg) - [`IsPID(O): RngOrd → BoolElt`](operation.md#function-ispid-rngord) - [`IsUFD(O): RngOrd → BoolElt`](operation.md#function-isufd-rngord) - [`IsPrincipalIdealRing(O): RngOrd → BoolElt`](operation.md#function-isprincipalidealring-rngord) - [`IsDomain(R): FldAlg → BoolElt`](operation.md#function-isdomain-fldalg) - [`F ne L: FldAlg, FldAlg → BoolElt`](operation.md#operation-op-ne-fldalg-fldalg) - [`O ne N: RngOrd, RngOrd → BoolElt`](operation.md#operation-op-ne-rngord-rngord) - [`O subset P: RngOrd, RngOrd → BoolElt`](operation.md#operation-op-subset-rngord-rngord) - [`K subset L: FldAlg, FldAlg → BoolElt`](operation.md#operation-op-subset-fldalg-fldalg) - [`K subset L: FldRat, FldAlg → BoolElt`](operation.md#operation-op-subset-fldrat-fldalg) - [`HasComplexConjugate(K): FldAlg → BoolElt, Map`](operation.md#function-rngord-hascomplexconjugate) - [`ComplexConjugate(x): FldAlgElt → FldAlgElt`](operation.md#function-complexconjugate-fldalgelt) - [Order Predicates](operation.md#order-predicates) - [`IsEquationOrder(O): RngOrd → BoolElt`](operation.md#function-isequationorder-rngord) - [`IsMaximal(O): RngOrd → BoolElt`](operation.md#function-ismaximal-rngord) - [`IsAbsoluteOrder(O): RngOrd → BoolElt`](operation.md#function-isabsoluteorder-rngord) - [`IsWildlyRamified(O): RngOrd → BoolElt`](operation.md#function-iswildlyramified-rngord) - [`IsTamelyRamified(O): RngOrd → BoolElt`](operation.md#function-istamelyramified-rngord) - [`IsUnramified(O): RngOrd → BoolElt`](operation.md#function-isunramified-rngord) - [Field Predicates](operation.md#field-predicates) - [`IsIsomorphic(F, L): FldAlg, FldAlg → BoolElt, Map`](operation.md#function-isisomorphic-fldalg-fldalg) - [`IsSubfield(F, L): FldAlg, FldAlg → BoolElt, Map`](operation.md#function-issubfield-fldalg-fldalg) - [`IsSubfield(F, L): FldRat, FldAlg → BoolElt, Map`](operation.md#function-issubfield-fldrat-fldalg) - [`IsNormal(F): FldAlg → BoolElt`](operation.md#function-isnormal-fldalg) - [`IsAbelian(F): FldAlg → BoolElt`](operation.md#function-isabelian-fldalg) - [`IsCyclic(F): FldAlg → BoolElt`](operation.md#function-iscyclic-fldalg) - [`IsAbsoluteField(K): FldAlg → BoolElt`](operation.md#function-isabsolutefield-fldalg) - [`IsWildlyRamified(K): FldAlg → BoolElt`](operation.md#function-iswildlyramified-fldalg) - [`IsTamelyRamified(K): FldAlg → BoolElt`](operation.md#function-istamelyramified-fldalg) - [`IsUnramified(K): FldAlg → BoolElt`](operation.md#function-isunramified-fldalg) - [`IsQuadratic(K): FldAlg → BoolElt, FldQuad`](operation.md#function-isquadratic-fldalg) - [`IsTotallyReal(K): FldAlg → BoolElt`](operation.md#function-istotallyreal-fldalg) - [Setting Properties of Orders](operation.md#setting-properties-of-orders) - [`SetOrderMaximal(O, b): RngOrd, BoolElt`](operation.md#function-setordermaximal-rngord-boolelt) - [`SetOrderTorsionUnit(O, e, r): RngOrd, RngOrdElt, RngIntElt`](operation.md#function-setordertorsionunit-rngord-rngordelt-rngintelt) - [`SetOrderUnitsAreFundamental(O): RngOrd`](operation.md#function-rngord-setorderunitsarefundamental) - [Element Operations](operation-element.md) - [Parent and Category](operation-element.md#parent-and-category) - [`Parent(a): FldAlgElt → FldAlg`](operation-element.md#function-parent-fldalgelt) - [`Parent(w): RngOrdElt → RngOrd`](operation-element.md#function-parent-rngordelt) - [`Category(a): FldAlgElt → Cat`](operation-element.md#function-category-fldalgelt) - [`Category(w): RngOrdElt → Cat`](operation-element.md#function-category-rngordelt) - [Arithmetic](operation-element.md#arithmetic) - [`+ a: FldAlgElt → FldAlgElt`](operation-element.md#operation-operation-fldalgelt-fldalgelt) - [`+ w: RngOrdElt → RngOrdElt`](operation-element.md#operation-operation-rngordelt-rngordelt) - [`- a: FldAlgElt → FldAlgElt`](operation-element.md#operation-operation-fldalgelt-fldalgelt-2) - [`- w: RngOrdElt → RngOrdElt`](operation-element.md#operation-operation-rngordelt-rngordelt-2) - [`a + b: FldAlgElt, FldAlgElt → FldAlgElt`](operation-element.md#operation-op-plus-fldalgelt-fldalgelt) - [`w + v: RngOrdElt, RngOrdElt → RngOrdElt`](operation-element.md#operation-op-plus-rngordelt-rngordelt) - [`a - b: FldAlgElt, FldAlgElt → FldAlgElt`](operation-element.md#operation-op-minus-fldalgelt-fldalgelt) - [`w - v: RngOrdElt, RngOrdElt → RngOrdElt`](operation-element.md#operation-op-minus-rngordelt-rngordelt) - [`a * b: FldAlgElt, FldAlgElt → FldAlgElt`](operation-element.md#operation-op-times-fldalgelt-fldalgelt) - [`w * v: RngOrdElt, RngOrdElt → RngOrdElt`](operation-element.md#operation-op-times-rngordelt-rngordelt) - [`a / b: FldAlgElt, FldAlgElt → FldAlgElt`](operation-element.md#operation-op-div-fldalgelt-fldalgelt) - [`w / v: RngOrdElt, RngOrdElt → FldOrdElt`](operation-element.md#operation-op-div-rngordelt-rngordelt) - [`a ^ k: FldAlgElt, RngIntElt → FldAlgElt`](operation-element.md#operation-op-pow-fldalgelt-rngintelt) - [`w ^ k: RngOrdElt, RngIntElt → RngOrdElt`](operation-element.md#operation-op-pow-rngordelt-rngintelt) - [`w div v: RngOrdElt, RngOrdElt → RngOrdElt`](operation-element.md#operation-operation-div-rngordelt-rngordelt-rngordelt) - [`Modexp(a, n, m): RngOrdElt, RngIntElt, RngIntElt → RngOrdElt`](operation-element.md#function-modexp-rngordelt-rngintelt-rngintelt) - [`Sqrt(a): RngOrdElt → RngOrdElt`](operation-element.md#function-sqrt-rngordelt) - [`Sqrt(a): FldAlgElt → FldAlgElt`](operation-element.md#function-sqrt-fldalgelt) - [`SquareRoot(a): RngOrdElt → RngOrdElt`](operation-element.md#function-squareroot-rngordelt) - [`SquareRoot(a): FldAlgElt → FldAlgElt`](operation-element.md#function-squareroot-fldalgelt) - [`Root(a, n): RngOrdElt, RngIntElt → RngOrdElt`](operation-element.md#function-root-rngordelt-rngintelt) - [`Root(a, n): FldAlgElt, RngIntElt → FldAlgElt`](operation-element.md#function-root-fldalgelt-rngintelt) - [`IsPower(a, k): FldAlgElt, RngIntElt → BoolElt, FldAlgElt`](operation-element.md#function-ispower-fldalgelt-rngintelt) - [`IsPower(a, k): RngOrdElt, RngIntElt → BoolElt, RngOrdElt`](operation-element.md#function-ispower-rngordelt-rngintelt) - [`IsSquare(a): FldAlgElt → BoolElt, FldAlgElt`](operation-element.md#function-issquare-fldalgelt) - [`IsSquare(a): RngOrdElt → BoolElt, RngOrdElt`](operation-element.md#function-issquare-rngordelt) - [`Denominator(a): FldAlgElt → RngIntElt`](operation-element.md#function-denominator-fldalgelt) - [`Denominator(a): RngOrdElt → RngIntElt`](operation-element.md#function-denominator-rngordelt) - [`Numerator(a): FldAlgElt → RngIntElt`](operation-element.md#function-numerator-fldalgelt) - [`Qround(E, M): FldAlgElt, RngIntElt → FldAlgElt`](operation-element.md#function-qround-fldalgelt-rngintelt) - [Equality and Membership](operation-element.md#equality-and-membership) - [`a eq b: FldAlgElt, FldAlgElt → BoolElt`](operation-element.md#operation-op-eq-fldalgelt-fldalgelt) - [`w eq v: RngOrdElt, RngOrdElt → BoolElt`](operation-element.md#operation-op-eq-rngordelt-rngordelt) - [`a ne b: FldAlgElt, FldAlgElt → BoolElt`](operation-element.md#operation-op-ne-fldalgelt-fldalgelt) - [`w ne v: RngOrdElt, RngOrdElt → BoolElt`](operation-element.md#operation-op-ne-rngordelt-rngordelt) - [`a in F: FldAlgElt, FldAlg → BoolElt`](operation-element.md#operation-op-in-fldalgelt-fldalg) - [`w in O: RngOrdElt, RngOrd → BoolElt`](operation-element.md#operation-op-in-rngordelt-rngord) - [Predicates on Elements](operation-element.md#predicates-on-elements) - [`IsIntegral(a): FldAlgElt → BoolElt`](operation-element.md#function-isintegral-fldalgelt) - [`IsIntegral(a): RngOrdElt → BoolElt`](operation-element.md#function-isintegral-rngordelt) - [`IsPrimitive(a): FldAlgElt → BoolElt`](operation-element.md#function-isprimitive-fldalgelt) - [`IsPrimitive(a): RngOrdElt → BoolElt`](operation-element.md#function-isprimitive-rngordelt) - [`IsTorsionUnit(w): RngOrdElt → BoolElt`](operation-element.md#function-istorsionunit-rngordelt) - [`IsPower(w, n): RngOrdElt, RngIntElt → BoolElt, RngOrdElt`](operation-element.md#function-ispower-rngordelt-rngintelt-2) - [`IsTotallyPositive(a): RngOrdElt → BoolElt`](operation-element.md#function-istotallypositive-rngordelt) - [`IsTotallyPositive(a): FldAlgElt → BoolElt`](operation-element.md#function-istotallypositive-fldalgelt) - [`IsZero(a): FldAlgElt → BoolElt`](operation-element.md#function-iszero-fldalgelt) - [`IsZero(w): RngOrdElt → BoolElt`](operation-element.md#function-iszero-rngordelt) - [`IsOne(a): FldAlgElt → BoolElt`](operation-element.md#function-isone-fldalgelt) - [`IsOne(w): RngOrdElt → BoolElt`](operation-element.md#function-isone-rngordelt) - [`IsMinusOne(a): FldAlgElt → BoolElt`](operation-element.md#function-isminusone-fldalgelt) - [`IsMinusOne(w): RngOrdElt → BoolElt`](operation-element.md#function-isminusone-rngordelt) - [`IsUnit(a): FldAlgElt → BoolElt`](operation-element.md#function-isunit-fldalgelt) - [`IsUnit(w): RngOrdElt → BoolElt`](operation-element.md#function-isunit-rngordelt) - [`IsNilpotent(a): FldAlgElt → BoolElt`](operation-element.md#function-isnilpotent-fldalgelt) - [`IsNilpotent(a): RngOrdElt → BoolElt`](operation-element.md#function-isnilpotent-rngordelt) - [`IsIdempotent(a): FldAlgElt → BoolElt`](operation-element.md#function-isidempotent-fldalgelt) - [`IsIdempotent(a): RngOrdElt → BoolElt`](operation-element.md#function-isidempotent-rngordelt) - [`IsZeroDivisor(a): FldAlgElt → BoolElt`](operation-element.md#function-iszerodivisor-fldalgelt) - [`IsZeroDivisor(a): RngOrdElt → BoolElt`](operation-element.md#function-iszerodivisor-rngordelt) - [`IsRegular(a): FldAlgElt → BoolElt`](operation-element.md#function-isregular-fldalgelt) - [`IsRegular(a): RngOrdElt → BoolElt`](operation-element.md#function-isregular-rngordelt) - [`IsIrreducible(a): FldAlgElt → BoolElt`](operation-element.md#function-isirreducible-fldalgelt) - [`IsIrreducible(a): RngOrdElt → BoolElt`](operation-element.md#function-isirreducible-rngordelt) - [`IsPrime(a): FldAlgElt → BoolElt`](operation-element.md#function-isprime-fldalgelt) - [`IsPrime(a): RngOrdElt → BoolElt`](operation-element.md#function-isprime-rngordelt) - [Field Generators](operation-element.md#field-generators) - [`K . 1: FldNum → FldNumElt`](operation-element.md#operation-operation-fldnum-fldnumelt) - [`PrimitiveElement(K): FldNum → FldNumElt`](operation-element.md#function-primitiveelement-fldnum) - [`PrimitiveElement(F): FldOrd → FldOrdElt`](operation-element.md#function-primitiveelement-fldord) - [`PrimitiveElement(O): RngOrd → RngOrdElt`](operation-element.md#function-primitiveelement-rngord) - [`Generators(K): FldAlg → [FldAlgElt]`](operation-element.md#function-generators-fldalg) - [`Generators(K, k): FldAlg, FldAlg → [FldAlgElt]`](operation-element.md#function-generators-fldalg-fldalg) - [`Generators(K, Q): FldAlg, FldRat → [FldAlgElt]`](operation-element.md#function-generators-fldalg-fldrat) - [Real and Complex Embeddings](operation-element.md#real-and-complex-embeddings) - [`Conjugates(a): FldAlgElt → [ FldComElt ]`](operation-element.md#function-conjugates-fldalgelt) - [`Conjugates(a): RngOrdElt → [ FldComElt ]`](operation-element.md#function-conjugates-rngordelt) - [`Conjugate(a, l): FldAlgElt, RngIntElt → FldReElt`](operation-element.md#function-conjugate-fldalgelt-rngintelt) - [`Conjugate(a, l): RngOrdElt, RngIntElt → FldReElt`](operation-element.md#function-conjugate-rngordelt-rngintelt) - [`Conjugate(a, l): FldAlgElt, [RngIntElt] → FldReElt`](operation-element.md#function-conjugate-fldalgelt-rngintelt-2) - [`AbsoluteValues(a): FldAlgElt → [FldReElt]`](operation-element.md#function-absolutevalues-fldalgelt) - [`AbsoluteValues(a): RngOrdElt → [FldReElt]`](operation-element.md#function-absolutevalues-rngordelt) - [`Logs(a): FldAlgElt → [FldReElt]`](operation-element.md#function-logs-fldalgelt) - [`Logs(a): RngOrdElt → [FldReElt]`](operation-element.md#function-logs-rngordelt) - [`InfinitePlaces(K): FldAlg → [PlcNumElt]`](operation-element.md#function-infiniteplaces-fldalg) - [`InfinitePlaces(O): RngOrd → [PlcNumElt]`](operation-element.md#function-infiniteplaces-rngord) - [`Evaluate(x, p): FldAlgElt, PlcNumElt → RngElt`](operation-element.md#function-evaluate-fldalgelt-plcnumelt) - [`Evaluate(x, p): RngOrdElt, PlcNumElt → RngElt`](operation-element.md#function-evaluate-rngordelt-plcnumelt) - [`RealEmbeddings(a): FldAlgElt → []`](operation-element.md#function-realembeddings-fldalgelt) - [`RealEmbeddings(a): RngOrdElt → []`](operation-element.md#function-realembeddings-rngordelt) - [`MinkowskiLattice(O): RngOrd → Lat, Map`](operation-element.md#function-minkowskilattice-rngord) - [`Lattice(O): RngOrd → Lat, Map`](operation-element.md#function-lattice-rngord) - [`MinkowskiLattice(I): RngOrdIdl → Lat, Map`](operation-element.md#function-minkowskilattice-rngordidl) - [`Lattice(I): RngOrdIdl → Lat, Map`](operation-element.md#function-lattice-rngordidl) - [`MinkowskiSpace(F): FldAlg → Lat, Map`](operation-element.md#function-minkowskispace-fldalg) - [Heights](operation-element.md#heights) - [`AbsoluteLogarithmicHeight(a): FldAlgElt → FldReElt`](operation-element.md#function-absolutelogarithmicheight-fldalgelt) - [`AbsoluteLogarithmicHeight(a): RngOrdElt → FldReElt`](operation-element.md#function-absolutelogarithmicheight-rngordelt) - [`CoefficientHeight(E): RngOrdElt → RngIntElt`](operation-element.md#function-coefficientheight-rngordelt) - [`CoefficientHeight(E): FldAlgElt → RngIntElt`](operation-element.md#function-coefficientheight-fldalgelt) - [`CoefficientLength(E): RngOrdElt → RngIntElt`](operation-element.md#function-coefficientlength-rngordelt) - [`CoefficientLength(E): FldAlgElt → RngIntElt`](operation-element.md#function-coefficientlength-fldalgelt) - [`Length(a): FldAlgElt → FldReElt`](operation-element.md#function-rngord-length) - [`Length(a): RngOrdElt → FldReElt`](operation-element.md#function-length-rngordelt) - [`Example: Discriminant`](operation-element.md#example-ex-19f6ce) - [Norm, Trace, and Minimal Polynomial](operation-element.md#norm-trace-and-minimal-polynomial) - [`Norm(a): FldAlgElt → FldAlgElt`](operation-element.md#function-norm-fldalgelt) - [`Norm(a): FldAlgElt → FldRatElt`](operation-element.md#function-rngord-norm) - [`Norm(a, R): FldAlgElt, Rng → RngElt`](operation-element.md#function-norm-fldalgelt-rng) - [`Norm(a): RngOrdElt → RngOrdElt`](operation-element.md#function-norm-rngordelt) - [`Norm(a): RngOrdElt → RngIntElt`](operation-element.md#function-norm-rngordelt-2) - [`Norm(a, R): RngOrdElt, Rng → RngElt`](operation-element.md#function-norm-rngordelt-rng) - [`AbsoluteNorm(a): FldAlgElt → FldRatElt`](operation-element.md#function-absolutenorm-fldalgelt) - [`NormAbs(a): FldAlgElt → FldRatElt`](operation-element.md#function-normabs-fldalgelt) - [`AbsoluteNorm(a): RngOrdElt → FldRatElt`](operation-element.md#function-absolutenorm-rngordelt) - [`NormAbs(a): RngOrdElt → FldRatElt`](operation-element.md#function-normabs-rngordelt) - [`Trace(a): FldAlgElt → FldAlgElt`](operation-element.md#function-trace-fldalgelt) - [`Trace(a): FldAlgElt → FldRatElt`](operation-element.md#function-rngord-trace) - [`Trace(a, R): FldAlgElt, Rng → RngElt`](operation-element.md#function-trace-fldalgelt-rng) - [`Trace(a): RngOrdElt → RngOrdElt`](operation-element.md#function-trace-rngordelt) - [`Trace(a): RngOrdElt → RngIntElt`](operation-element.md#function-trace-rngordelt-2) - [`Trace(a, R): RngOrdElt, Rng → RngElt`](operation-element.md#function-trace-rngordelt-rng) - [`AbsoluteTrace(a): FldAlgElt → FldRatElt`](operation-element.md#function-absolutetrace-fldalgelt) - [`TraceAbs(a): FldAlgElt → FldRatElt`](operation-element.md#function-traceabs-fldalgelt) - [`AbsoluteTrace(a): RngOrdElt → FldRatElt`](operation-element.md#function-absolutetrace-rngordelt) - [`TraceAbs(a): RngOrdElt → FldRatElt`](operation-element.md#function-traceabs-rngordelt) - [`CharacteristicPolynomial(a): FldAlgElt → RngUPolElt`](operation-element.md#function-characteristicpolynomial-fldalgelt) - [`CharacteristicPolynomial(a, R): FldAlgElt, Rng → RngUPolElt`](operation-element.md#function-characteristicpolynomial-fldalgelt-rng) - [`CharacteristicPolynomial(a): RngOrdElt → RngUPolElt`](operation-element.md#function-characteristicpolynomial-rngordelt) - [`CharacteristicPolynomial(a, R): RngOrdElt, Rng → RngUPolElt`](operation-element.md#function-characteristicpolynomial-rngordelt-rng) - [`AbsoluteCharacteristicPolynomial(a): FldAlgElt → RngUPolElt`](operation-element.md#function-absolutecharacteristicpolynomial-fldalgelt) - [`AbsoluteCharacteristicPolynomial(a): RngOrdElt → RngUPolElt`](operation-element.md#function-absolutecharacteristicpolynomial-rngordelt) - [`MinimalPolynomial(a): FldAlgElt → RngUPolElt`](operation-element.md#function-minimalpolynomial-fldalgelt) - [`MinimalPolynomial(a, R): FldAlgElt, Rng → RngUPolElt`](operation-element.md#function-minimalpolynomial-fldalgelt-rng) - [`MinimalPolynomial(a): RngOrdElt → RngUPolElt`](operation-element.md#function-minimalpolynomial-rngordelt) - [`MinimalPolynomial(a, R): RngOrdElt, Rng → RngUPolElt`](operation-element.md#function-minimalpolynomial-rngordelt-rng) - [`AbsoluteMinimalPolynomial(a): FldAlgElt → RngUPolElt`](operation-element.md#function-absoluteminimalpolynomial-fldalgelt) - [`AbsoluteMinimalPolynomial(a): RngOrdElt → RngUPolElt`](operation-element.md#function-absoluteminimalpolynomial-rngordelt) - [`RepresentationMatrix(a): FldAlgElt → AlgMatElt`](operation-element.md#function-representationmatrix-fldalgelt) - [`RepresentationMatrix(a, R): FldAlgElt, Rng → AlgMatElt`](operation-element.md#function-representationmatrix-fldalgelt-rng) - [`RepresentationMatrix(a): RngOrdElt → AlgMatElt`](operation-element.md#function-representationmatrix-rngordelt) - [`RepresentationMatrix(a, R): RngOrdElt, Rng → AlgMatElt`](operation-element.md#function-representationmatrix-rngordelt-rng) - [`AbsoluteRepresentationMatrix(a): FldAlgElt → AlgMatElt`](operation-element.md#function-absoluterepresentationmatrix-fldalgelt) - [`AbsoluteRepresentationMatrix(a): RngOrdElt → AlgMatElt`](operation-element.md#function-absoluterepresentationmatrix-rngordelt) - [`Example: Norms Etc`](operation-element.md#example-ex-68a043) - [The Quadratic Defect](operation-element.md#the-quadratic-defect) - [`QuadraticDefect(a, p): RngElt, RngOrdIdl → RngIntElt`](operation-element.md#function-quadraticdefect-rngelt-rngordidl) - [`QuadraticDefect(a, p): FldRatElt, RngIntElt → RngIntElt`](operation-element.md#function-quadraticdefect-fldratelt-rngintelt) - [`QuadraticDefect(a, p): RngIntElt, RngIntElt → RngIntElt`](operation-element.md#function-quadraticdefect-rngintelt-rngintelt) - [`QuadraticDefect(a, p): FldRatElt, RngInt → RngIntElt`](operation-element.md#function-quadraticdefect-fldratelt-rngint) - [`QuadraticDefect(a, p): RngIntElt, RngInt → RngIntElt`](operation-element.md#function-quadraticdefect-rngintelt-rngint) - [`RelativeQuadraticDefect(a, p): RngElt, RngOrdIdl → RngIntElt`](operation-element.md#function-relativequadraticdefect-rngelt-rngordidl) - [`RelativeQuadraticDefect(a, p): FldRatElt, RngIntElt → RngIntElt`](operation-element.md#function-relativequadraticdefect-fldratelt-rngintelt) - [`RelativeQuadraticDefect(a, p): RngIntElt, RngIntElt → RngIntElt`](operation-element.md#function-relativequadraticdefect-rngintelt-rngintelt) - [`RelativeQuadraticDefect(a, p): FldRatElt, RngInt → RngIntElt`](operation-element.md#function-relativequadraticdefect-fldratelt-rngint) - [`RelativeQuadraticDefect(a, p): RngIntElt, RngInt → RngIntElt`](operation-element.md#function-relativequadraticdefect-rngintelt-rngint) - [`IsLocalSquare(a, p): RngElt, RngOrdIdl → BoolElt`](operation-element.md#function-islocalsquare-rngelt-rngordidl) - [`IsLocalSquare(a, p): FldRatElt, RngIntElt → BoolElt`](operation-element.md#function-islocalsquare-fldratelt-rngintelt) - [`IsLocalSquare(a, p): RngIntElt, RngIntElt → BoolElt`](operation-element.md#function-islocalsquare-rngintelt-rngintelt) - [`IsLocalSquare(a, p): FldRatElt, RngInt → BoolElt`](operation-element.md#function-islocalsquare-fldratelt-rngint) - [`IsLocalSquare(a, p): RngIntElt, RngInt → BoolElt`](operation-element.md#function-islocalsquare-rngintelt-rngint) - [`LocalMultiplicativeGroupModSquares(p): RngOrdIdl → ModFld, Map`](operation-element.md#function-localmultiplicativegroupmodsquares-rngordidl) - [`LocalMultiplicativeGroupModSquares(p): RngIntElt → ModFld, Map`](operation-element.md#function-localmultiplicativegroupmodsquares-rngintelt) - [`UnitSquareClassReps(p): RngOrdIdl → SeqEnum`](operation-element.md#function-unitsquareclassreps-rngordidl) - [`UnitSquareClassReps(p): RngIntElt → SeqEnum`](operation-element.md#function-unitsquareclassreps-rngintelt) - [`NiceUnitSquareClassRepresentative(u, p): RngElt, RngOrdIdl → RngElt`](operation-element.md#function-niceunitsquareclassrepresentative-rngelt-rngordidl) - [`NiceUnitSquareClassRepresentative(u, p): RngElt, RngIntElt → RngElt`](operation-element.md#function-niceunitsquareclassrepresentative-rngelt-rngintelt) - [Other Functions](operation-element.md#other-functions) - [`ElementToSequence(a): FldAlgElt → [ FldAlgElt ]`](operation-element.md#function-elementtosequence-fldalgelt) - [`Eltseq(a): FldAlgElt → [ FldAlgElt ]`](operation-element.md#function-eltseq-fldalgelt) - [`ElementToSequence(a): RngOrdElt → [ FldOrdElt ]`](operation-element.md#function-elementtosequence-rngordelt) - [`Eltseq(a): RngOrdElt → [ FldOrdElt ]`](operation-element.md#function-rngord-eltseq) - [`Eltseq(E, k): FldAlgElt, FldAlg → [RngElt]`](operation-element.md#function-eltseq-fldalgelt-fldalg) - [`Eltseq(E, k): FldAlgElt, Rng → [RngElt]`](operation-element.md#function-eltseq-fldalgelt-rng) - [`Flat(e): FldAlgElt → [FldRatElt]`](operation-element.md#function-flat-fldalgelt) - [`a[i]: FldAlgElt, RngIntElt → FldRatElt`](operation-element.md#literal-literal-a-i-fldalgelt-rngintelt-fldratelt) - [`a[i]: FldAlgElt, RngIntElt → FldAlgElt`](operation-element.md#literal-literal-a-i-fldalgelt-rngintelt-fldalgelt) - [`a[i]: RngOrdElt, RngIntElt → FldRatElt`](operation-element.md#literal-literal-a-i-rngordelt-rngintelt-fldratelt) - [`a[i]: RngOrdElt, RngIntElt → FldOrdElt`](operation-element.md#literal-literal-a-i-rngordelt-rngintelt-fldordelt) - [`ProductRepresentation(a): RngOrdElt → [ RngOrdElt ], [ RngIntElt ]`](operation-element.md#function-productrepresentation-rngordelt) - [`ProductRepresentation(a): FldAlgElt → [ FldAlgElt ], [ RngIntElt ]`](operation-element.md#function-productrepresentation-fldalgelt) - [`ProductRepresentation(P, E): [ FldAlgElt ], [ RngIntElt ] → FldAlgElt`](operation-element.md#function-productrepresentation-fldalgelt-rngintelt) - [`PowerProduct(P, E): [FldAlgElt], [RngIntElt] → FldAlgElt`](operation-element.md#function-powerproduct-fldalgelt-rngintelt) - [`Valuation(w, I): RngOrdElt, RngOrdIdl → RngIntElt`](operation-element.md#function-valuation-rngordelt-rngordidl) - [`Valuation(w, I): FldAlgElt, RngOrdIdl → RngIntElt`](operation-element.md#function-valuation-fldalgelt-rngordidl) - [`Decomposition(a): RngOrdElt → SeqEnum[]`](operation-element.md#function-decomposition-rngordelt) - [`Decomposition(a): FldOrdElt → SeqEnum[]`](operation-element.md#function-decomposition-fldordelt) - [`Divisors(a): RngOrdElt → SeqEnum[RngOrdElt]`](operation-element.md#function-divisors-rngordelt) - [`Index(a): RngOrdElt → RngIntElt`](operation-element.md#function-index-rngordelt) - [`Different(a): RngOrdElt → RngOrdElt`](operation-element.md#function-different-rngordelt) - [`DegreeOnePrimeIdeals(O, B): RngOrd, RngIntElt → [ RngOrdIdl ]`](operation-element.md#function-degreeoneprimeideals-rngord-rngintelt) - [Ideal Class Groups](ideal-class-group.md) - [`SetPrintClassGroupWarnings(b): BoolElt`](ideal-class-group.md#function-setprintclassgroupwarnings-boolelt) - [`ClassGroup(O: parameters): RngOrd → GrpAb, Map`](ideal-class-group.md#function-rngord-classgroup) - [`ClassGroup(K: parameters): FldNum → GrpAb, Map`](ideal-class-group.md#function-classgroup-fldnum) - [`RingClassGroup(O): RngOrd → GrpAb, Map`](ideal-class-group.md#function-ringclassgroup-rngord) - [`PicardGroup(O): RngOrd → GrpAb, Map`](ideal-class-group.md#function-picardgroup-rngord) - [`ConditionalClassGroup(O): RngOrd → GrpAb, Map`](ideal-class-group.md#function-conditionalclassgroup-rngord) - [`ConditionalClassGroup(K): FldNum → GrpAb, Map`](ideal-class-group.md#function-conditionalclassgroup-fldnum) - [`ClassGroupPrimeRepresentatives(O, I): RngOrd, RngOrdIdl → Map`](ideal-class-group.md#function-classgroupprimerepresentatives-rngord-rngordidl) - [`ClassNumber(O: parameters): RngOrd → RngIntElt`](ideal-class-group.md#function-classnumber-rngord) - [`ClassNumber(K: parameters): FldNum → RngIntElt`](ideal-class-group.md#function-classnumber-fldnum) - [`MinkowskiBound(K): FldNum → RngIntElt`](ideal-class-group.md#function-minkowskibound-fldnum) - [`MinkowskiBound(O): RngOrd → RngIntElt`](ideal-class-group.md#function-minkowskibound-rngord) - [`BachBound(K): FldNum → RngIntElt`](ideal-class-group.md#function-bachbound-fldnum) - [`BachBound(O): RngOrd → RngIntElt`](ideal-class-group.md#function-bachbound-rngord) - [`GRHBound(K): FldNum → RngIntElt`](ideal-class-group.md#function-grhbound-fldnum) - [`GRHBound(O): RngOrd → RngIntElt`](ideal-class-group.md#function-grhbound-rngord) - [`FactorBasisVerify(O, a, b): RngOrd, RngIntElt, RngIntElt`](ideal-class-group.md#function-factorbasisverify-rngord-rngintelt-rngintelt) - [Class Group Internals](ideal-class-group.md#class-group-internals) - [`EulerProduct(O, B): RngOrd, RngIntElt → FldReElt`](ideal-class-group.md#function-eulerproduct-rngord-rngintelt) - [`ResidueGRH(O, B): RngOrd, RngIntElt → FldReElt`](ideal-class-group.md#function-residuegrh-rngord-rngintelt) - [`ResidueGRHbound(O, e): RngOrd, FldReElt → FldReElt`](ideal-class-group.md#function-residuegrhbound-rngord-fldreelt) - [`FactorBasis(K, B): FldNum, RngIntElt → [ RngOrdIdl ]`](ideal-class-group.md#function-factorbasis-fldnum-rngintelt) - [`FactorBasis(O, B): RngOrd, RngIntElt → [ RngOrdIdl ]`](ideal-class-group.md#function-factorbasis-rngord-rngintelt) - [`FactorBasis(O): RngOrd → [ RngOrdIdl ], Integer`](ideal-class-group.md#function-factorbasis-rngord) - [`RelationMatrix(O): RngOrd → ModHomElt`](ideal-class-group.md#function-relationmatrix-rngord) - [`Relations(O): RngOrd → ModHomElt`](ideal-class-group.md#function-relations-rngord) - [`ClassGroupCyclicFactorGenerators(O): RngOrd → ModHomElt`](ideal-class-group.md#function-classgroupcyclicfactorgenerators-rngord) - [`Example: Class Group`](ideal-class-group.md#example-ex-92eeba) - [Setting the Class Group Bounds](ideal-class-group.md#setting-the-class-group-bounds) - [`SetClassGroupBounds(string): MonStgElt`](ideal-class-group.md#function-setclassgroupbounds-monstgelt) - [`SetClassGroupBounds(n): RngIntElt`](ideal-class-group.md#function-setclassgroupbounds-rngintelt) - [`Example: Class Group Bounds`](ideal-class-group.md#example-ex-3fbfbd) - [Class Group Map Caching](ideal-class-group.md#class-group-map-caching) - [`ClassGroupGetUseMemory(O): RngOrd → BoolElt`](ideal-class-group.md#function-classgroupgetusememory-rngord) - [`ClassGroupSetUseMemory(O, f): RngOrd, BoolElt`](ideal-class-group.md#function-classgroupsetusememory-rngord-boolelt) - [Unit Groups](unit-group.md) - [`UnitRank(O): RngOrd → RngIntElt`](unit-group.md#function-unitrank-rngord-2) - [`UnitRank(K): FldAlg → RngIntElt`](unit-group.md#function-unitrank-fldalg) - [`TorsionUnitGroup(O): RngOrd → GrpAb, Map`](unit-group.md#function-torsionunitgroup-rngord) - [`TorsionUnitGroup(K): FldNum → GrpAb, Map`](unit-group.md#function-torsionunitgroup-fldnum) - [`UnitGroup(O): RngOrd → GrpAb, Map`](unit-group.md#function-rngord-unitgroup) - [`MultiplicativeGroup(O): RngOrd → GrpAb, Map`](unit-group.md#function-multiplicativegroup-rngord) - [`UnitGroup(K): FldNum → GrpAb, Map`](unit-group.md#function-unitgroup-fldnum) - [`MultiplicativeGroup(K): FldNum → GrpAb, Map`](unit-group.md#function-multiplicativegroup-fldnum) - [`IndependentUnits(O): RngOrd → GrpAb, Map`](unit-group.md#function-independentunits-rngord) - [`IndependentUnits(K): FldNum → GrpAb, Map`](unit-group.md#function-independentunits-fldnum) - [`pFundamentalUnits(O, p): RngOrd, RngIntElt → GrpAb, Map`](unit-group.md#function-pfundamentalunits-rngord-rngintelt) - [`pFundamentalUnits(K, p): FldNum, RngIntElt → GrpAb, Map`](unit-group.md#function-pfundamentalunits-fldnum-rngintelt) - [`UnitGroupAsSubgroup(O): RngOrd → GrpAb`](unit-group.md#function-unitgroupassubgroup-rngord) - [`MergeUnits(K, a): FldNum, FldNumElt → BoolElt`](unit-group.md#function-mergeunits-fldnum-fldnumelt) - [`MergeUnits(O, a): RngOrd, RngOrdElt → BoolElt`](unit-group.md#function-mergeunits-rngord-rngordelt) - [`Example: Unit Group`](unit-group.md#example-ex-98001b) - [`IsExceptionalUnit(u): RngOrdElt → BoolElt`](unit-group.md#function-isexceptionalunit-rngordelt) - [`ExceptionalUnitOrbit(u): RngOrdElt → [ RngOrdElt ]`](unit-group.md#function-exceptionalunitorbit-rngordelt) - [`ExceptionalUnits(O): RngOrd → [ RngOrdElt ]`](unit-group.md#function-exceptionalunits-rngord) - [`UnitsWithSigns(O, oo, Signs): RngOrd, [ PlcNumElt ], [ RngInt ] → [ RngOrdElt ]`](unit-group.md#function-unitswithsigns-rngord-plcnumelt-rngint) - [`UnitsWithSigns(K, oo, Signs): FldAlg, [ PlcNumElt ], [ RngInt ] → [ FldAlgElt ]`](unit-group.md#function-unitswithsigns-fldalg-plcnumelt-rngint) - [`UnitsWithSigns(O, x): RngOrd, RngElt → [ RngOrdElt ]`](unit-group.md#function-unitswithsigns-rngord-rngelt) - [`UnitsWithSigns(K, x): FldAlg, RngElt → [ FldAlgElt ]`](unit-group.md#function-unitswithsigns-fldalg-rngelt) - [`UnitsWithSigns(x): RngOrdElt → [ RngOrdElt ]`](unit-group.md#function-unitswithsigns-rngordelt) - [`UnitsWithSigns(x): FldAlgElt → [ FldAlgElt ]`](unit-group.md#function-unitswithsigns-fldalgelt) - [`HasTotallyPositiveGenerator(I): RngOrdFracIdl → BoolElt, [ RngOrdElt ]`](unit-group.md#function-hastotallypositivegenerator-rngordfracidl) - [`TraceMinimalCone(nf): FldNum → TorCon`](unit-group.md#function-traceminimalcone-fldnum) - [Diophantine Equations](equation.md) - [Norm Equations](equation.md#norm-equations) - [`NormEquation(O, m): RngOrd, RngIntElt → BoolElt, [ RngOrdElt ]`](equation.md#function-normequation-rngord-rngintelt) - [`NormEquation(O, m): RngOrd, RngOrdElt → BoolElt, [ RngOrdElt ]`](equation.md#function-normequation-rngord-rngordelt) - [`NormEquation(F, m): FldAlg, RngIntElt → BoolElt, [ FldAlgElt ]`](equation.md#function-rngord-normequation-fld) - [`NormEquation(F, m): FldAlg, FldAlgElt → BoolElt, [ FldAlgElt ]`](equation.md#function-normequation-fldalg-fldalgelt) - [`NormEquation(F, m): FldAlg, FldRatElt → BoolElt, [ FldAlgElt ]`](equation.md#function-normequation-fldalg-fldratelt) - [`NormEquation(m, N): RngElt, Map → BoolElt, RngElt`](equation.md#function-normequation-rngelt-map) - [`IntegralNormEquation(a, N, O): RngElt, Map, RngOrd → BoolElt, [RngOrdElt]`](equation.md#function-integralnormequation-rngelt-map-rngord) - [`SimNEQ(K, e, f): FldNum, FldNumElt, FldNumElt → BoolElt, [FldNumElt]`](equation.md#function-simneq-fldnum-fldnumelt-fldnumelt) - [`Example: Norm Equation`](equation.md#example-ex-a68dc5) - [Thue Equations](equation.md#thue-equations) - [`ThueSystem(f): RngUPolElt → Thue`](equation.md#function-thuesystem-rngupolelt) - [`ThueSystem(O): RngOrd → Thue`](equation.md#function-thuesystem-rngord) - [`Evaluate(t, a, b): Thue, RngIntElt, RngIntElt → RngIntElt`](equation.md#function-evaluate-thue-rngintelt-rngintelt) - [`Evaluate(t, S): Thue, [ RngIntElt ] → RngIntElt`](equation.md#function-evaluate-thue-rngintelt) - [`Solutions(t, a): Thue, RngIntElt → [ [ RngIntElt, RngIntElt ] ]`](equation.md#function-solutions-thue-rngintelt) - [`Example: thue`](equation.md#example-ex-cbfc14) - [Unit Equations](equation.md#unit-equations) - [`UnitEquation(a, b, c): FldNumElt, FldNumElt, FldNumElt → [ ModHomElt ]`](equation.md#function-unitequation-fldnumelt-fldnumelt-fldnumelt) - [`Example: uniteq`](equation.md#example-ex-5cb066) - [Index Form Equations](equation.md#index-form-equations) - [`IndexFormEquation(O, k): RngOrd, RngIntElt → [ RngOrdElt ]`](equation.md#function-indexformequation-rngord-rngintelt) - [`Example: Index Form`](equation.md#example-ex-a3384f) - [Ideals and Quotients](ideal.md) - [Creation of Ideals in Orders](ideal.md#creation-of-ideals-in-orders) - [`x * O: RngElt, RngOrd → RngOrdFracIdl`](ideal.md#operation-op-times-rngelt-rngord) - [`O * x: RngElt, RngOrd → RngOrdFracIdl`](ideal.md#operation-op-times-rngelt-rngord-2) - [`F !! I: RngOrd, RngInt → RngOrdFracIdl`](ideal.md#operation-op-rngord-rngint) - [`F !! I: FldOrd, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#operation-op-fldord-rngordfracidl) - [`F !! I: FldRat, RngOrdFracIdl → RngIntFracIdl`](ideal.md#operation-op-fldrat-rngordfracidl) - [`F !! I: FldRat, RngIntFracIdl → RngIntFracIdl`](ideal.md#operation-op-fldrat-rngintfracidl) - [`F !! I: FldOrd, RngIntFracIdl → RngOrdFracIdl`](ideal.md#operation-op-fldord-rngintfracidl) - [`O !! I: RngOrd, RngOrdFracIdl → RngOrdIdl`](ideal.md#operation-op-rngord-rngordfracidl) - [`O !! I: RngInt, RngInt → RngInt`](ideal.md#operation-op-rngint-rngint) - [`ideal< O | a₁, a₂, ... , aₘ >: RngOrd, RngElt, ..., RngElt → RngOrdFracIdl`](ideal.md#constructor-constructor-ideal-rngord-rngelt-rngelt-rngordfracidl) - [`ideal: RngOrd, RngIntElt → RngOrdIdl`](ideal.md#constructor-constructor-ideal-rngord-rngintelt-rngordidl) - [`ideal: RngOrd, AlgMatElt, RngIntElt → RngOrdFracIdl`](ideal.md#constructor-constructor-ideal-rngord-algmatelt-rngintelt-rngordfracidl) - [`ideal: RngOrd, ModDed, RngIntElt → RngOrdFracIdl`](ideal.md#constructor-constructor-ideal-rngord-modded-rngintelt-rngordfracidl) - [`ideal: RngOrd, AlgMatElt, RngOrdFracIdl, ..., RngOrdFracIdl → RngOrdFracIdl`](ideal.md#constructor-constructor-ideal-rngord-algmatelt-rngordfracidl-rngordfracidl-rngordfracidl) - [`ideal: RngOrd, AlgMatElt, [RngOrdFracIdl] → RngOrdFracIdl`](ideal.md#constructor-constructor-ideal-rngord-algmatelt-rngordfracidl-rngordfracidl) - [`Example: Ideals`](ideal.md#example-ex-dc3704) - [`FractionalIdeal(x): FldRatElt → RngIntFracIdl`](ideal.md#function-fractionalideal-fldratelt) - [`FractionalIdeal(x): RngIntElt → RngIntFracIdl`](ideal.md#function-fractionalideal-rngintelt) - [`FractionalIdeal(x): [ FldRatElt ] → RngIntFracIdl`](ideal.md#function-fractionalideal-fldratelt-2) - [`FractionalIdeal(x): [ RngIntElt ] → RngIntFracIdl`](ideal.md#function-fractionalideal-rngintelt-2) - [`FractionalIdeal(I): RngInt → RngIntFracIdl`](ideal.md#function-fractionalideal-rngint) - [Invariants](ideal.md#invariants) - [`Order(I): RngOrdFracIdl → RngOrd`](ideal.md#function-order-rngordfracidl) - [`Order(I): RngIntFracIdl → RngInt`](ideal.md#function-order-rngintfracidl) - [`Denominator(I): RngOrdFracIdl → RngIntElt`](ideal.md#function-denominator-rngordfracidl) - [`Denominator(I): RngIntFracIdl → RngIntElt`](ideal.md#function-denominator-rngintfracidl) - [`PrimitiveElement(I): RngOrdIdl → RngOrdElt`](ideal.md#function-primitiveelement-rngordidl) - [`PrimitiveElement(I): RngOrdFracIdl → FldOrdElt`](ideal.md#function-primitiveelement-rngordfracidl) - [`UniformizingElement(P): RngOrdIdl → RngOrdElt`](ideal.md#function-uniformizingelement-rngordidl) - [`UniformizingElement(P): RngInt → RngIntElt`](ideal.md#function-uniformizingelement-rngint) - [`Index(O, I): RngOrd, RngOrdIdl → RngIntElt`](ideal.md#function-index-rngord-rngordidl) - [`Norm(I): RngOrdIdl → RngIntElt`](ideal.md#function-norm-rngordidl) - [`Norm(I): RngOrdFracIdl → FldRatElt`](ideal.md#function-norm-rngordfracidl) - [`Norm(I): RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-norm-rngordfracidl-2) - [`Norm(I): RngIntFracIdl → RngIntElt`](ideal.md#function-norm-rngintfracidl) - [`MinimalInteger(I): RngOrdIdl → RngElt`](ideal.md#function-minimalinteger-rngordidl) - [`Minimum(I): RngOrdFracIdl → RngElt`](ideal.md#function-minimum-rngordfracidl) - [`Min(I): RngOrdFracIdl → RngElt`](ideal.md#function-min-rngordfracidl) - [`AbsoluteNorm(I): RngOrdIdl → RngIntElt`](ideal.md#function-absolutenorm-rngordidl) - [`AbsoluteNorm(I): RngOrdFracIdl → FldRatElt`](ideal.md#function-absolutenorm-rngordfracidl) - [`NormAbs(I): RngOrdIdl → RngIntElt`](ideal.md#function-normabs-rngordidl) - [`NormAbs(I): RngOrdFracIdl → FldRatElt`](ideal.md#function-normabs-rngordfracidl) - [`CoefficientHeight(I): RngOrdIdl → RngIntElt`](ideal.md#function-coefficientheight-rngordidl) - [`CoefficientHeight(I): RngOrdFracIdl → RngIntElt`](ideal.md#function-coefficientheight-rngordfracidl) - [`CoefficientLength(I): RngOrdIdl → RngIntElt`](ideal.md#function-coefficientlength-rngordidl) - [`CoefficientLength(I): RngOrdFracIdl → RngIntElt`](ideal.md#function-coefficientlength-rngordfracidl) - [`RamificationIndex(I, p): RngOrdIdl, RngIntElt → RngIntElt`](ideal.md#function-ramificationindex-rngordidl-rngintelt) - [`RamificationDegree(I, p): RngOrdIdl, RngIntElt → RngIntElt`](ideal.md#function-ramificationdegree-rngordidl-rngintelt) - [`RamificationDegree(I): RngOrdIdl → RngIntElt`](ideal.md#function-ramificationdegree-rngordidl) - [`RamificationIndex(I): RngOrdIdl → RngIntElt`](ideal.md#function-ramificationindex-rngordidl) - [`AbsoluteRamificationDegree(I): RngOrdIdl → RngIntElt`](ideal.md#function-absoluteramificationdegree-rngordidl) - [`AbsoluteRamificationIndex(I): RndOrdIdl → RngIntElt`](ideal.md#function-absoluteramificationindex-rndordidl) - [`ResidueClassField(O, I): RngOrd, RngOrdIdl → FldFin, Map`](ideal.md#function-residueclassfield-rngord-rngordidl) - [`ResidueClassField(I): RngOrdIdl → FldFin, Map`](ideal.md#function-residueclassfield-rngordidl) - [`Degree(I): RngOrdIdl → RngIntElt`](ideal.md#function-degree-rngordidl) - [`InertiaDegree(I): RngOrdIdl → RngIntElt`](ideal.md#function-inertiadegree-rngordidl) - [`AbsoluteInertiaDegree(I): RngOrdIdl → RngIntElt`](ideal.md#function-absoluteinertiadegree-rngordidl) - [`AbsoluteInertiaIndex(I): RngOrdIdl → RngIntElt`](ideal.md#function-absoluteinertiaindex-rngordidl) - [`Valuation(I, p): RngOrdFracIdl, RngOrdIdl → RngIntElt`](ideal.md#function-valuation-rngordfracidl-rngordidl) - [`Valuation(I, p): RngIntFracIdl, RngInt → RngIntElt`](ideal.md#function-valuation-rngintfracidl-rngint) - [`Content(I): RngOrdFracIdl → RngIntElt`](ideal.md#function-content-rngordfracidl) - [`Example: Ideal Invar`](ideal.md#example-ex-f01e3a) - [Basis Representation](ideal.md#basis-representation) - [`Basis(I): RngOrdIdl → [RngOrdElt]`](ideal.md#function-basis-rngordidl) - [`Basis(I): RngOrdFracIdl → [FldOrdElt]`](ideal.md#function-basis-rngordfracidl) - [`Basis(I): RngIntFracIdl → [FldRatElt]`](ideal.md#function-basis-rngintfracidl) - [`Basis(I, R): RngOrdFracIdl, Rng → [RngElt]`](ideal.md#function-basis-rngordfracidl-rng) - [`BasisMatrix(I): RngOrdFracIdl → MtrxSpcElt`](ideal.md#function-basismatrix-rngordfracidl) - [`TransformationMatrix(I): RngOrdFracIdl → MtrxSpcElt, RngIntElt`](ideal.md#function-transformationmatrix-rngordfracidl) - [`CoefficientIdeals(I): RngOrdFracIdl → [RngOrdFracIdl]`](ideal.md#function-coefficientideals-rngordfracidl) - [`Example: Ideal Basis`](ideal.md#example-ex-49f139) - [`Module(I): RngOrdFracIdl → ModDed, Map`](ideal.md#function-module-rngordfracidl) - [Two–Element Presentations](ideal.md#two-element-presentations) - [`Generators(I): RngOrdIdl → [ RngOrdElt ]`](ideal.md#function-generators-rngordidl) - [`Generators(I): RngOrdFracIdl → [ FldOrdElt ]`](ideal.md#function-generators-rngordfracidl) - [`Generators(I): RngIntFracIdl → [ FldRatElt ]`](ideal.md#function-generators-rngintfracidl) - [`TwoElement(I): RngOrdFracIdl → FldOrdElt, FldOrdElt`](ideal.md#function-twoelement-rngordfracidl) - [`TwoElementNormal(I): RngOrdIdl → RngOrdElt, RngOrdElt, RngIntElt`](ideal.md#function-twoelementnormal-rngordidl) - [`Example: Ideal Two`](ideal.md#example-ex-4c86cb) - [Standard Names](ideal.md#standard-names) - [`LMFDBLabel(I): RngOrdIdl → MonStgElg`](ideal.md#function-lmfdblabel-rngordidl) - [`LMFDBIdeal(K, s): FldNum, MonStgElt → RngOrdIdl`](ideal.md#function-lmfdbideal-fldnum-monstgelt) - [Predicates on Ideals](ideal.md#predicates-on-ideals) - [`I eq J: RngOrdFracIdl, RngOrdFracIdl → BoolElt`](ideal.md#operation-op-eq-rngordfracidl-rngordfracidl) - [`I ne J: RngOrdFracIdl, RngOrdFracIdl → BoolElt`](ideal.md#operation-op-ne-rngordfracidl-rngordfracidl) - [`x in I: FldOrdElt, RngOrdFracIdl → BoolElt`](ideal.md#operation-op-in-fldordelt-rngordfracidl) - [`x in I: RngOrdElt, RngOrdFracIdl → BoolElt`](ideal.md#operation-op-in-rngordelt-rngordfracidl) - [`x notin I: FldOrdElt, RngOrdFracIdl → BoolElt`](ideal.md#operation-op-notin-fldordelt-rngordfracidl) - [`x notin I: RngOrdElt, RngOrdFracIdl → BoolElt`](ideal.md#operation-op-notin-rngordelt-rngordfracidl) - [`I subset J: RngOrdIdl, RngOrdIdl → BoolElt`](ideal.md#operation-op-subset-rngordidl-rngordidl) - [`IsIntegral(I): RngOrdFracIdl → BoolElt`](ideal.md#function-isintegral-rngordfracidl) - [`IsIntegral(I): RngIntFracIdl → BoolElt`](ideal.md#function-isintegral-rngintfracidl) - [`IsZero(I): RngOrdFracIdl → BoolElt`](ideal.md#function-iszero-rngordfracidl) - [`IsOne(I): RngOrdIdl → BoolElt`](ideal.md#function-isone-rngordidl) - [`IsOne(I): RngOrdFracIdl → BoolElt`](ideal.md#function-isone-rngordfracidl) - [`IsOne(I): RngIntFracIdl → BoolElt`](ideal.md#function-isone-rngintfracidl) - [`IsPrime(I): RngOrdIdl → BoolElt, RngOrdIdl`](ideal.md#function-isprime-rngordidl) - [`IsPrime(I): RngOrdFracIdl → BoolElt`](ideal.md#function-isprime-rngordfracidl) - [`IsPrincipal(I): RngOrdFracIdl → BoolElt, FldOrdElt`](ideal.md#function-isprincipal-rngordfracidl) - [`IsRamified(P): RngOrdIdl → BoolElt`](ideal.md#function-isramified-rngordidl) - [`IsRamified(P, O): RngOrdIdl, RngOrd → BoolElt`](ideal.md#function-isramified-rngordidl-rngord) - [`IsRamified(P, O): RngIntElt, RngOrd → BoolElt`](ideal.md#function-isramified-rngintelt-rngord) - [`IsRamified(P, O): RngInt, RngOrd → BoolElt`](ideal.md#function-isramified-rngint-rngord) - [`IsSquarefree(I): RngOrdIdl → BoolElt`](ideal.md#function-issquarefree-rngordidl) - [`IsSquarefree(I): RngInt → BoolElt`](ideal.md#function-issquarefree-rngint) - [`IsTotallyRamified(P): RngOrdIdl → BoolElt`](ideal.md#function-istotallyramified-rngordidl) - [`IsTotallyRamified(P, O): RngOrdIdl, RngOrd → BoolElt`](ideal.md#function-istotallyramified-rngordidl-rngord) - [`IsTotallyRamified(P, O): RngIntElt, RngOrd → BoolElt`](ideal.md#function-istotallyramified-rngintelt-rngord) - [`IsTotallyRamified(K): FldAlg → BoolElt`](ideal.md#function-istotallyramified-fldalg) - [`IsTotallyRamified(O): RngOrd → BoolElt`](ideal.md#function-istotallyramified-rngord) - [`IsWildlyRamified(P): RngOrdIdl → BoolElt`](ideal.md#function-iswildlyramified-rngordidl) - [`IsWildlyRamified(P, O): RngOrdIdl, RngOrd → BoolElt`](ideal.md#function-iswildlyramified-rngordidl-rngord) - [`IsWildlyRamified(P, O): RngIntElt, RngOrd → BoolElt`](ideal.md#function-iswildlyramified-rngintelt-rngord) - [`IsTamelyRamified(P): RngOrdIdl → BoolElt`](ideal.md#function-istamelyramified-rngordidl) - [`IsTamelyRamified(P, O): RngOrdIdl, RngOrd → BoolElt`](ideal.md#function-istamelyramified-rngordidl-rngord) - [`IsTamelyRamified(P, O): RngIntElt, RngOrd → BoolElt`](ideal.md#function-istamelyramified-rngintelt-rngord) - [`IsUnramified(P): RngOrdIdl → BoolElt`](ideal.md#function-isunramified-rngordidl) - [`IsUnramified(P, O): RngOrdIdl, RngOrd → BoolElt`](ideal.md#function-isunramified-rngordidl-rngord) - [`IsUnramified(P, O): RngIntElt, RngOrd → BoolElt`](ideal.md#function-isunramified-rngintelt-rngord) - [`IsInert(P): RngOrdIdl → BoolElt`](ideal.md#function-isinert-rngordidl) - [`IsInert(P, O): RngOrdIdl, RngOrd → BoolElt`](ideal.md#function-isinert-rngordidl-rngord) - [`IsInert(P, O): RngIntElt, RngOrd → BoolElt`](ideal.md#function-isinert-rngintelt-rngord) - [`IsInert(P, O): RngInt, RngOrd → BoolElt`](ideal.md#function-isinert-rngint-rngord) - [`IsSplit(P): RngOrdIdl → BoolElt`](ideal.md#function-issplit-rngordidl) - [`IsSplit(P, O): RngOrdIdl, RngOrd → BoolElt`](ideal.md#function-issplit-rngordidl-rngord) - [`IsSplit(P, O): RngIntElt, RngOrd → BoolElt`](ideal.md#function-issplit-rngintelt-rngord) - [`IsSplit(P, O): RngInt, RngOrd → BoolElt`](ideal.md#function-issplit-rngint-rngord) - [`IsTotallySplit(P): RngOrdIdl → BoolElt`](ideal.md#function-istotallysplit-rngordidl) - [`IsTotallySplit(P, O): RngOrdIdl, RngOrd → BoolElt`](ideal.md#function-istotallysplit-rngordidl-rngord) - [`IsTotallySplit(P, O): RngIntElt, RngOrd → BoolElt`](ideal.md#function-istotallysplit-rngintelt-rngord) - [`IsTotallySplit(P, O): RngInt, RngOrd → BoolElt`](ideal.md#function-istotallysplit-rngint-rngord) - [Ideal Arithmetic](ideal.md#ideal-arithmetic) - [`I * J: RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#operation-op-times-rngordfracidl-rngordfracidl) - [`I * J: RngIntFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#operation-op-times-rngintfracidl-rngordfracidl) - [`I * J: RngIntFracIdl, RngIntFracIdl → RngIntFracIdl`](ideal.md#operation-op-times-rngintfracidl-rngintfracidl) - [`x * I: RngElt, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#operation-op-times-rngelt-rngordfracidl) - [`x * I: FldRatElt, RngIntFracIdl → RngIntFracIdl`](ideal.md#operation-op-times-fldratelt-rngintfracidl) - [`x * I: RngIntElt, RngIntFracIdl → RngIntFracIdl`](ideal.md#operation-op-times-rngintelt-rngintfracidl) - [`I * x: RngOrdFracIdl, RngElt → RngOrdFracIdl`](ideal.md#operation-op-times-rngordfracidl-rngelt) - [`&* L: [RngOrdFracIdl] → RngOrdFracIdl`](ideal.md#operation-operation-rngordfracidl-rngordfracidl) - [`&* L: { RngOrdFracIdl } → RngOrdFracIdl`](ideal.md#operation-operation-rngordfracidl-rngordfracidl-2) - [`I div J: RngOrdIdl, RngOrdIdl → RngOrdIdl`](ideal.md#operation-operation-div-rngordidl-rngordidl-rngordidl) - [`I div J: RngOrdFracIdl, RngOrdFracIdl → RngOrdIdl`](ideal.md#operation-operation-div-rngordfracidl-rngordfracidl-rngordidl) - [`I div J: RngInt, RngInt → RngInt`](ideal.md#operation-operation-div-rngint-rngint-rngint) - [`I / J: RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#operation-op-div-rngordfracidl-rngordfracidl) - [`I / x: RngOrdFracIdl, RngElt → RngOrdFracIdl`](ideal.md#operation-op-div-rngordfracidl-rngelt) - [`I + J: RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#operation-op-plus-rngordfracidl-rngordfracidl) - [`I + J: RngIntFracIdl, RngIntFracIdl → RngIntFracIdl`](ideal.md#operation-op-plus-rngintfracidl-rngintfracidl) - [`I + J: RngIntFracIdl, RngInt → RngIntFracIdl`](ideal.md#operation-op-plus-rngintfracidl-rngint) - [`I ^ k: RngOrdFracIdl, RngIntElt → RngOrdFracIdl`](ideal.md#operation-op-pow-rngordfracidl-rngintelt) - [`I ^ k: RngIntFracIdl, RngIntElt → RngIntFracIdl`](ideal.md#operation-op-pow-rngintfracidl-rngintelt) - [`I eq J: RngOrdFracIdl, RngOrdFracIdl → BoolElt`](ideal.md#operation-op-eq-rngordfracidl-rngordfracidl-2) - [`I eq J: RngIntFracIdl, RngIntFracIdl → BoolElt`](ideal.md#operation-op-eq-rngintfracidl-rngintfracidl) - [`I eq J: RngIntFracIdl, RngInt → BoolElt`](ideal.md#operation-op-eq-rngintfracidl-rngint) - [`I subset J: RngOrdIdl, RngOrdIdl → BoolElt`](ideal.md#operation-op-subset-rngordidl-rngordidl-2) - [`E in I: RngOrdElt, RngOrdIdl → BoolElt`](ideal.md#operation-op-in-rngordelt-rngordidl) - [`E in I: RngOrdElt, RngOrdFracIdl → BoolElt`](ideal.md#operation-op-in-rngordelt-rngordfracidl-2) - [`E in I: FldAlgElt, RngOrdFracIdl → BoolElt`](ideal.md#operation-op-in-fldalgelt-rngordfracidl) - [`LCM(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-lcm-rngordfracidl-rngordfracidl) - [`Lcm(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-lcm-rngordfracidl-rngordfracidl-2) - [`LeastCommonMultiple(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-leastcommonmultiple-rngordfracidl-rngordfracidl) - [`GCD(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-gcd-rngordfracidl-rngordfracidl) - [`Gcd(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-gcd-rngordfracidl-rngordfracidl-2) - [`GreatestCommonDivisor(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-greatestcommondivisor-rngordfracidl-rngordfracidl) - [`Content(M): Mtrx → RngOrdFracIdl`](ideal.md#function-content-mtrx) - [`I meet J: RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#operation-op-meet-rngordfracidl-rngordfracidl) - [`&meet S: [RngOrdFracIdl] → RngOrdFracIdl`](ideal.md#operation-operation-meet-rngordfracidl-rngordfracidl) - [`&meet S: { RngOrdFracIdl } → RngOrdFracIdl`](ideal.md#operation-operation-meet-rngordfracidl-rngordfracidl-2) - [`I meet R: RngOrdFracIdl, Rng → Any`](ideal.md#operation-op-meet-rngordfracidl-rng) - [`R meet I: Rng, RngOrdFracIdl → Any`](ideal.md#operation-op-meet-rng-rngordfracidl) - [`a mod I: RngOrdElt, RngOrdIdl → RngOrdElt`](ideal.md#operation-operation-mod-rngordelt-rngordidl-rngordelt) - [`InverseMod(E, M): RngOrdElt, RngIntElt → RngOrdElt`](ideal.md#function-inversemod-rngordelt-rngintelt) - [`InverseMod(E, M): RngOrdElt, RngOrdIdl → RngOrdElt`](ideal.md#function-inversemod-rngordelt-rngordidl) - [`Modinv(E, M): RngOrdElt, RngOrdIdl → RngOrdElt`](ideal.md#function-modinv-rngordelt-rngordidl) - [`Modinv(E, M): RngOrdElt, RngIntElt → RngOrdElt`](ideal.md#function-modinv-rngordelt-rngintelt) - [`ColonIdeal(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-colonideal-rngordfracidl-rngordfracidl) - [`IdealQuotient(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-idealquotient-rngordfracidl-rngordfracidl) - [`Example: colon`](ideal.md#example-ex-9af5d2) - [`IntegralSplit(I): RngOrdFracIdl → RngOrdIdl, RngElt`](ideal.md#function-integralsplit-rngordfracidl) - [`Different(I): RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-different-rngordfracidl) - [`Codifferent(I): RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-codifferent-rngordfracidl) - [Roots of Ideals](ideal.md#roots-of-ideals) - [`Root(I, k): RngOrdFracIdl, RngIntElt → RngOrdFracIdl`](ideal.md#function-root-rngordfracidl-rngintelt) - [`IsPower(I, k): RngOrdFracIdl, RngIntElt → BoolElt, RngOrdFracIdl`](ideal.md#function-ispower-rngordfracidl-rngintelt) - [`SquareRoot(I): RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-squareroot-rngordfracidl) - [`Sqrt(I): RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-sqrt-rngordfracidl) - [`IsSquare(I): RngOrdFracIdl → BoolElt, RngOrdFracIdl`](ideal.md#function-issquare-rngordfracidl) - [Factorization and Primes](ideal.md#factorization-and-primes) - [`Decomposition(O, p): RngOrd, RngIntElt → []`](ideal.md#function-decomposition-rngord-rngintelt) - [`Decomposition(O, p): RngOrd, RngOrdIdl → [ ]`](ideal.md#function-decomposition-rngord-rngordidl) - [`DecompositionType(O, p): RngOrd, RngIntElt → []`](ideal.md#function-decompositiontype-rngord-rngintelt) - [`DecompositionType(O, p): FldAlg, RngIntElt → []`](ideal.md#function-decompositiontype-fldalg-rngintelt) - [`DecompositionType(O, p): FldAlg, RngOrdIdl → []`](ideal.md#function-decompositiontype-fldalg-rngordidl) - [`DecompositionType(O, p): RngOrd, RngOrdIdl → [ ]`](ideal.md#function-decompositiontype-rngord-rngordidl) - [`Factorization(I): RngOrdFracIdl → []`](ideal.md#function-factorization-rngordfracidl) - [`Factorisation(I): RngOrdFracIdl → []`](ideal.md#function-factorisation-rngordfracidl) - [`Factorization(I): RngIntFracIdl → []`](ideal.md#function-factorization-rngintfracidl) - [`Factorisation(I): RngIntFracIdl → []`](ideal.md#function-factorisation-rngintfracidl) - [`Example: Non Maximal Fact`](ideal.md#example-ex-4f7652) - [`Divisors(I): RngOrdIdl → []`](ideal.md#function-divisors-rngordidl) - [`Support(I): RngOrdFracIdl → {RngOrdIdl}`](ideal.md#function-support-rngordfracidl) - [`Support(I): RngIntFracIdl → {RngInt}`](ideal.md#function-support-rngintfracidl) - [`Support(L): [RngOrdFracIdl] → {RngOrdIdl}`](ideal.md#function-support-rngordfracidl-2) - [`Support(L): [FldAlgElt] → {RngOrdIdl}`](ideal.md#function-support-fldalgelt) - [`CoprimeBasis(L): [RngOrdFracIdl] → RngOrdIdl`](ideal.md#function-coprimebasis-rngordfracidl) - [`CoprimeBasisInsert(~L, I): [RngOrdIdl], RngOrdFracIdl`](ideal.md#function-coprimebasisinsert-rngordidl-rngordfracidl-ref) - [`PowerProduct(B, E): [RngOrdFracIdl], [RngIntElt] → RngOrdFracIdl`](ideal.md#function-powerproduct-rngordfracidl-rngintelt) - [Other Ideal Operations](ideal.md#other-ideal-operations) - [`ChineseRemainderTheorem(I1, I2, e1, e2): RngOrdIdl, RngOrdIdl, RngOrdElt, RngOrdElt → RngOrdElt`](ideal.md#function-chineseremaindertheorem-rngordidl-rngordidl-rngordelt-rngordelt) - [`ChineseRemainderTheorem(X, M): SeqEnum[RngOrdElt], SeqEnum[RngOrdIdl] → RngOrdElt`](ideal.md#function-chineseremaindertheorem-seqenum-rngordelt-seqenum-rngordidl) - [`CRT(I1, I2, e1, e2): RngOrdIdl, RngOrdIdl, RngOrdElt, RngOrdElt → RngOrdElt`](ideal.md#function-crt-rngordidl-rngordidl-rngordelt-rngordelt) - [`CRT(X, M): SeqEnum[RngOrdElt], SeqEnum[RngOrdIdl] → RngOrdElt`](ideal.md#function-crt-seqenum-rngordelt-seqenum-rngordidl) - [`CRT(I1, L1, e1, L2): RngOrdIdl, [RngIntElt], RngOrdElt, [RngIntElt] → RngOrdElt`](ideal.md#function-rngord-chineseremaindertheoreminf) - [`ChineseRemainderTheorem(I1, L1, e1, L2): RngOrdIdl, [RngIntElt], RngOrdElt, [RngIntElt] → RngOrdElt`](ideal.md#function-chineseremaindertheorem-rngordidl-rngintelt-rngordelt-rngintelt) - [`WeakApproximation(I, V): [RngOrdIdl], [RngIntElt] → FldOrdElt`](ideal.md#function-weakapproximation-rngordidl-rngintelt) - [`WeakApproximation(I, V): [RngInt], [RngIntElt] → FldRatElt`](ideal.md#function-weakapproximation-rngint-rngintelt) - [`Idempotents(I, J): RngOrdIdl, RngOrdIdl → BoolElt, RngOrdElt, RngOrdElt`](ideal.md#function-idempotents-rngordidl-rngordidl) - [`CoprimeRepresentative(I, J): RngOrdIdl, RngOrdIdl → FldOrdElt`](ideal.md#function-coprimerepresentative-rngordidl-rngordidl) - [`MakeCoprime(I, J): RngOrdIdl, RngOrdIdl → FldOrdElt`](ideal.md#function-makecoprime-rngordidl-rngordidl) - [`IdealsUpTo(B, O): RngIntElt, RngOrd → [RngOrdIdl]`](ideal.md#function-idealsupto-rngintelt-rngord) - [`IdealsUpTo(B, K): RngIntElt, FldNum → [RngOrdIdl]`](ideal.md#function-idealsupto-rngintelt-fldnum) - [`ClassRepresentative(I): RngOrdFracIdl → RngOrdFracIdl`](ideal.md#function-classrepresentative-rngordfracidl) - [`SUnitGroup(I): RngOrdFracIdl → GrpAb, Map`](ideal.md#function-sunitgroup-rngordfracidl) - [`SUnitGroup(S): [ RngOrdIdl ] → GrpAb, Map`](ideal.md#function-rngord-sunitgroup) - [`Example: S Units`](ideal.md#example-ex-062dcc) - [`SUnitAction(SU, Act, S): Map, Map, SeqEnum[RngOrdIdl] → Map`](ideal.md#function-sunitaction-map-map-seqenum-rngordidl) - [`SUnitAction(SU, Act, S): Map, SeqEnum[Map], SeqEnum[RngOrdIdl] → [Map]`](ideal.md#function-sunitaction-map-seqenum-map-seqenum-rngordidl) - [`SUnitDiscLog(SU, x, S): Map, FldAlgElt, SeqEnum[RngOrdIdl] → GrpAbElt`](ideal.md#function-sunitdisclog-map-fldalgelt-seqenum-rngordidl) - [`SUnitDiscLog(SU, L, S): Map, SeqEnum[FldAlgElt], SeqEnum[RngOrdIdl] → GrpAbElt`](ideal.md#function-sunitdisclog-map-seqenum-fldalgelt-seqenum-rngordidl) - [`Example: S-Units, advanced`](ideal.md#example-ex-95d956) - [Quotient Rings](ideal.md#quotient-rings) - [Operations on Quotient Rings](ideal.md#operations-on-quotient-rings) - [`quo< O | I >: RngOrd, RngOrdIdl → RngOrdRes`](ideal.md#constructor-constructor-quo-rngord-rngordidl-rngordres) - [`quo< O | M >: RngOrd, ModDed → RngOrdRes`](ideal.md#constructor-constructor-quo-rngord-modded-rngordres) - [`quo< O | M >: RngOrd, AlgMatElt → RngOrdRes`](ideal.md#constructor-constructor-quo-rngord-algmatelt-rngordres) - [`quo< O | S >: RngOrd, RngElt, ..., RngElt → RngOrdRes`](ideal.md#constructor-constructor-quo-rngord-rngelt-rngelt-rngordres) - [`quo< O | m >: RngOrd, RngIntElt → RngOrdRes`](ideal.md#constructor-constructor-quo-rngord-rngintelt-rngordres) - [`UnitGroup(OQ): RngOrdRes → GrpAb, Map`](ideal.md#function-unitgroup-rngordres) - [`MultiplicativeGroup(OQ): RngOrdRes → GrpAb, Map`](ideal.md#function-multiplicativegroup-rngordres) - [`Modulus(OQ): RngOrdRes → RngOrdIdl`](ideal.md#function-modulus-rngordres) - [`Example: quotient`](ideal.md#example-ex-d59aa1) - [Elements of Quotients](ideal.md#elements-of-quotients) - [`OQ ! a: RngOrdRes, Elt → RngOrdResElt`](ideal.md#operation-op-rngordres-elt) - [`Random(OQ): RngOrdRes → RngOrdResElt`](ideal.md#function-random-rngordres) - [`a mod I: RngOrdElt, RngOrdIdl → RngOrdElt`](ideal.md#operation-operation-mod-rngordelt-rngordidl-rngordelt-2) - [`a * b: RngOrdResElt, RngOrdResElt → RngOrdResElt`](ideal.md#operation-op-times-rngordreselt-rngordreselt) - [`a + b: RngOrdResElt, RngOrdResElt → RngOrdResElt`](ideal.md#operation-op-plus-rngordreselt-rngordreselt) - [`a - b: RngOrdResElt, RngOrdResElt → RngOrdResElt`](ideal.md#operation-op-minus-rngordreselt-rngordreselt) - [`a / b: RngOrdResElt, RngOrdResElt → RngOrdResElt`](ideal.md#operation-op-div-rngordreselt-rngordreselt) - [`- a: RngOrdResElt → RngOrdResElt`](ideal.md#operation-operation-rngordreselt-rngordreselt) - [`a ^ n: RngOrdResElt, RngIntElt → RngOrdResElt`](ideal.md#operation-op-pow-rngordreselt-rngintelt) - [`a eq b: RngOrdResElt, RngOrdResElt → BoolElt`](ideal.md#operation-op-eq-rngordreselt-rngordreselt) - [`a ne b: RngOrdResElt, RngOrdResElt → BoolElt`](ideal.md#operation-op-ne-rngordreselt-rngordreselt) - [`a div b: RngOrdResElt, RngOrdResElt → RngOrdResElt`](ideal.md#operation-operation-div-rngordreselt-rngordreselt-rngordreselt) - [`a mod b: RngOrdResElt, RngOrdResElt → RngOrdResElt`](ideal.md#operation-operation-mod-rngordreselt-rngordreselt-rngordreselt) - [`Quotrem(a, b): RngOrdResElt, RngOrdResElt → RngOrdResElt, RngOrdResElt`](ideal.md#function-quotrem-rngordreselt-rngordreselt) - [`Lcm(a, b): RngOrdResElt, RngOrdResElt → RngOrdResElt`](ideal.md#function-lcm-rngordreselt-rngordreselt) - [`Gcd(a, b): RngOrdResElt, RngOrdResElt → RngOrdResElt`](ideal.md#function-gcd-rngordreselt-rngordreselt) - [`XGcd(a, b): RngOrdResElt, RngOrdResElt → RngOrdResElt, RngOrdResElt, RngOrdResElt`](ideal.md#function-xgcd-rngordreselt-rngordreselt) - [`IsZero(a): RngOrdResElt → BoolElt`](ideal.md#function-iszero-rngordreselt) - [`IsOne(a): RngOrdResElt → BoolElt`](ideal.md#function-isone-rngordreselt) - [`IsMinusOne(a): RngOrdResElt → BoolElt`](ideal.md#function-isminusone-rngordreselt) - [`IsUnit(a): RngOrdResElt → BoolElt`](ideal.md#function-isunit-rngordreselt) - [`Eltseq(a): RngOrdResElt → []`](ideal.md#function-eltseq-rngordreselt) - [`ElementToSequence(a): RngOrdResElt → []`](ideal.md#function-elementtosequence-rngordreselt) - [`EuclideanNorm(a): RngOrdResElt → RngIntElt`](ideal.md#function-euclideannorm-rngordreselt) - [`Annihilator(a): RngOrdResElt → RngOrdResElt`](ideal.md#function-annihilator-rngordreselt) - [Reconstruction](ideal.md#reconstruction) - [`ReconstructionEnvironment(p, k): RngOrdIdl, RngIntElt → RngOrdRecoEnv`](ideal.md#function-reconstructionenvironment-rngordidl-rngintelt) - [`ReconstructionEnvironment(p, k): RngIntElt, RngIntElt → RngOrdRecoEnv`](ideal.md#function-reconstructionenvironment-rngintelt-rngintelt) - [`Reconstruct(x, R): RngOrdElt, RngOrdRecoEnv → RngOrdElt`](ideal.md#function-reconstruct-rngordelt-rngordrecoenv) - [`Reconstruct(x, R): RngIntElt, RngOrdRecoEnv → RngIntElt`](ideal.md#function-reconstruct-rngintelt-rngordrecoenv) - [`ChangePrecision(~R, k): RngOrdRecoEnv, RngIntElt`](ideal.md#function-changeprecision-rngordrecoenv-rngintelt-ref) - [`Example: Order Reco`](ideal.md#example-ex-c0c980) - [Places and Divisors](divisors.md) - [Creation of Structures](divisors.md#creation-of-structures) - [`Places(K): FldNum → PlcNum`](divisors.md#function-places-fldnum) - [`DivisorGroup(K): FldNum → DivNum`](divisors.md#function-divisorgroup-fldnum) - [Operations on Structures](divisors.md#operations-on-structures) - [`d1 eq d2: DivNum, DivNum → BoolElt`](divisors.md#operation-op-eq-divnum-divnum) - [`p1 eq p2: PlcNum, PlcNum → BoolElt`](divisors.md#operation-op-eq-plcnum-plcnum) - [`NumberField(P): PlcNum → FldNum`](divisors.md#function-numberfield-plcnum) - [`NumberField(D): DivNum → FldNum`](divisors.md#function-numberfield-divnum) - [Creation of Elements](divisors.md#creation-of-elements) - [`Place(I): RngOrdIdl → PlcNumElt`](divisors.md#function-place-rngordidl) - [`Decomposition(K, I): FldAlg, Infty → SeqEnum`](divisors.md#function-decomposition-fldalg-infty) - [`Decomposition(K, p): FldAlg, RngIntElt → SeqEnum`](divisors.md#function-decomposition-fldalg-rngintelt) - [`Decomposition(K, p): FldNum, PlcNumElt → SeqEnum`](divisors.md#function-decomposition-fldnum-plcnumelt) - [`Decomposition(m, p): Map[FldRat, FldAlg], RngIntElt → SeqEnum[]`](divisors.md#function-decomposition-map-fldrat-fldalg-rngintelt) - [`Decomposition(m, p): Map[FldAlg, FldAlg], PlcNumElt → SeqEnum[]`](divisors.md#function-decomposition-map-fldalg-fldalg-plcnumelt) - [`InfinitePlaces(K): FldAlg → [PlcNumElt]`](divisors.md#function-infiniteplaces-fldalg-2) - [`InfinitePlaces(O): RngOrd → [PlcNumElt]`](divisors.md#function-infiniteplaces-rngord-2) - [`RealPlaces(K): FldAlg → [PlcNumElt]`](divisors.md#function-realplaces-fldalg) - [`Divisor(pl): PlcNumElt → DivNumElt`](divisors.md#function-divisor-plcnumelt) - [`Divisor(I): RngOrdFracIdl → DivNumElt`](divisors.md#function-divisor-rngordfracidl) - [`Divisor(x): FldNumElt → DivNumElt`](divisors.md#function-divisor-fldnumelt) - [Arithmetic with Places and Divisors](divisors.md#arithmetic-with-places-and-divisors) - [`d1 + d2: DivNumElt, DivNumElt → DivNumElt`](divisors.md#operation-op-plus-divnumelt-divnumelt) - [`p + d: PlcNumElt, DivNumElt → DivNumElt`](divisors.md#operation-op-plus-plcnumelt-divnumelt) - [`d + p: DivNumElt, PlcNumElt → DivNumElt`](divisors.md#operation-op-plus-divnumelt-plcnumelt) - [`p1 + p2: PlcNumElt, PlcNumElt → DivNumElt`](divisors.md#operation-op-plus-plcnumelt-plcnumelt) - [`- p: PlcNumElt → DivNumElt`](divisors.md#operation-operation-plcnumelt-divnumelt) - [`- d: DivNumElt → DivNumElt`](divisors.md#operation-operation-divnumelt-divnumelt) - [`d - p: DivNumElt, PlcNumElt → DivNumElt`](divisors.md#operation-op-minus-divnumelt-plcnumelt) - [`p - d: PlcNumElt, DivNumElt → DivNumElt`](divisors.md#operation-op-minus-plcnumelt-divnumelt) - [`d1 - d2: DivNumElt, DivNumElt → DivNumElt`](divisors.md#operation-op-minus-divnumelt-divnumelt) - [`p1 - p2: PlcNumElt, PlcNumElt → DivNumElt`](divisors.md#operation-op-minus-plcnumelt-plcnumelt) - [`p * k: PlcNumElt, RngIntElt → DivNumElt`](divisors.md#operation-op-times-plcnumelt-rngintelt) - [`d * k: DivNumElt, RngIntElt → DivNumElt`](divisors.md#operation-op-times-divnumelt-rngintelt) - [`p div k: PlcNumElt, RngIntElt → DivNumElt`](divisors.md#operation-operation-div-plcnumelt-rngintelt-divnumelt) - [`d div k: DivNumElt, RngIntElt → DivNumElt`](divisors.md#operation-operation-div-divnumelt-rngintelt-divnumelt) - [Other Functions for Places and Divisors](divisors.md#other-functions-for-places-and-divisors) - [`Valuation(a, p): FldNumElt, PlcNumElt → RngElt`](divisors.md#function-valuation-fldnumelt-plcnumelt) - [`Valuation(a, p): RngElt, PlcNumElt → RngElt`](divisors.md#function-valuation-rngelt-plcnumelt) - [`Valuation(I, p): RngOrdFracIdl, PlcNumElt → RngElt`](divisors.md#function-valuation-rngordfracidl-plcnumelt) - [`Support(D): DivNumElt → SeqEnum, SeqEnum`](divisors.md#function-support-divnumelt) - [`Ideal(D): DivNumElt → RngOrdIdl`](divisors.md#function-ideal-divnumelt) - [`Ideal(D): PlcNumElt → RngOrdIdl`](divisors.md#function-ideal-plcnumelt) - [`IsFinite(p): PlcNumElt → BoolElt`](divisors.md#function-isfinite-plcnumelt) - [`IsInfinite(p): PlcNumElt → BoolElt, RngIntElt`](divisors.md#function-isinfinite-plcnumelt) - [`IsReal(p): PlcNumElt → BoolElt`](divisors.md#function-isreal-plcnumelt) - [`IsComplex(p): PlcNumElt → BoolElt`](divisors.md#function-iscomplex-plcnumelt) - [`Extends(P, p): PlcNumElt, PlcNumElt → BoolElt`](divisors.md#function-extends-plcnumelt-plcnumelt) - [`InertiaDegree(P): PlcNumElt → RngIntElt`](divisors.md#function-inertiadegree-plcnumelt) - [`Degree(P): PlcNumElt → RngIntElt`](divisors.md#function-degree-plcnumelt) - [`Degree(D): DivNumElt → RngElt`](divisors.md#function-degree-divnumelt) - [`NumberField(P): PlcNumElt → FldNum`](divisors.md#function-numberfield-plcnumelt) - [`NumberField(D): DivNumElt → FldNum`](divisors.md#function-numberfield-divnumelt) - [`ResidueClassField(P): PlcNumElt → Fld`](divisors.md#function-residueclassfield-plcnumelt) - [`UniformizingElement(P): PlcNumElt → FldNumElt`](divisors.md#function-uniformizingelement-plcnumelt) - [`LocalDegree(P): PlcNumElt → RngIntElt`](divisors.md#function-localdegree-plcnumelt) - [`RamificationIndex(P): PlcNumElt → RngIntElt`](divisors.md#function-ramificationindex-plcnumelt) - [`DecompositionGroup(P): PlcNumElt → GrpPerm`](divisors.md#function-decompositiongroup-plcnumelt) - [The Montes Algorithm](montes.md) - [`Montes(f, p): RngUPolElt, RngElt → SeqEnum, SeqEnum, RngIntElt`](montes.md#function-montes-rngupolelt-rngelt) - [`Example: Montes Eg 1`](montes.md#example-ex-25d08e) - [`Montes(K, p): FldArith, RngElt`](montes.md#function-montes-fldarith-rngelt) - [`Example: Montes Eg 2`](montes.md#example-ex-cb9d42) - [`SFL(P, s): OMIdl, RngIntElt`](montes.md#function-sfl-omidl-rngintelt) - [`Example: sfl`](montes.md#example-ex-e93eb7) - [`SetUseMontes(f): BoolElt`](montes.md#function-setusemontes-boolelt) - [`SetUseMontes(t, f): Cat, BoolElt`](montes.md#function-setusemontes-cat-boolelt) - [`GetUseMontes(t): Cat → BoolElt`](montes.md#function-getusemontes-cat) - [`SetVerbose("Montes", v): MonStgElt, RngIntElt`](montes.md#function-setverbose-monstgelt-rngintelt-2) - [Ideals in OM Representation](ideal-rep.md) - [`Ideal(I): OMIdl → RngOrdIdl`](ideal-rep.md#function-ideal-omidl) - [`OMRepresentation(I): RngFunOrdIdl → OMIdl`](ideal-rep.md#function-omrepresentation-rngfunordidl) - [`OMRepresentation(I): RngOrdFracIdl → OMIdl`](ideal-rep.md#function-omrepresentation-rngordfracidl) - [`OMRepresentation(L, S): FldArith, [FldArithElt] → OMIdl`](ideal-rep.md#function-omrepresentation-fldarith-fldarithelt) - [`OMRepresentation(L, a): FldArith, FldArithElt → OMIdl`](ideal-rep.md#function-omrepresentation-fldarith-fldarithelt-2) - [`OMRepresentation(L, a): FldArith, RngElt → OMIdl`](ideal-rep.md#function-omrepresentation-fldarith-rngelt) - [`Example: Om Rep Ideal`](ideal-rep.md#example-ex-738d2c) - [Ideal Arithmetic](ideal-rep.md#ideal-arithmetic) - [`I + J: OMIdl, OMIdl → OMIdl`](ideal-rep.md#operation-op-plus-omidl-omidl) - [`I * J: OMIdl, OMIdl → OMIdl`](ideal-rep.md#operation-op-times-omidl-omidl) - [`I / J: OMIdl, OMIdl → OMIdl`](ideal-rep.md#operation-op-div-omidl-omidl) - [`I ^ n: OMIdl, RngIntElt → OMIdl`](ideal-rep.md#operation-op-pow-omidl-rngintelt) - [`Example: Om Ideal Arith`](ideal-rep.md#example-ex-58ff3a) - [Ideal Predicates](ideal-rep.md#ideal-predicates) - [`IsOne(I): OMIdl → BoolElt`](ideal-rep.md#function-isone-omidl) - [`IsZero(I): OMIdl → BoolElt`](ideal-rep.md#function-iszero-omidl) - [`I eq J: OMIdl, OMIdl → BoolElt`](ideal-rep.md#operation-op-eq-omidl-omidl) - [`a in I: RngElt, OMIdl → BoolElt`](ideal-rep.md#operation-op-in-rngelt-omidl) - [`I subset J: OMIdl, OMIdl → BoolElt`](ideal-rep.md#operation-op-subset-omidl-omidl) - [`IsPrime(I): OMIdl → BoolElt`](ideal-rep.md#function-isprime-omidl) - [`IsIntegral(I): OMIdl → BoolElt`](ideal-rep.md#function-isintegral-omidl) - [Ideal Operations](ideal-op.md) - [`pIntegralBasis(I, p): OMIdl, RngElt → SeqEnum`](ideal-op.md#function-pintegralbasis-omidl-rngelt) - [`SIntegralBasis(I, S): OMIdl, SeqEnum → SeqEnum`](ideal-op.md#function-sintegralbasis-omidl-seqenum) - [`SIntegralBasis(I, S): RngOrdFracIdl, [RngIntElt] → SeqEnum`](ideal-op.md#function-sintegralbasis-rngordfracidl-rngintelt) - [`SIntegralBasis(I, S): RngFunOrdIdl, [RngUPolElt] → SeqEnum`](ideal-op.md#function-sintegralbasis-rngfunordidl-rngupolelt) - [`Basis(I): OMIdl → SeqEnum`](ideal-op.md#function-basis-omidl) - [`Example: Om Ideal Op`](ideal-op.md#example-ex-a203ed) - [`TwoElement(I): OMIdl → FldArithElt, FldArithElt`](ideal-op.md#function-twoelement-omidl) - [`Norm(I): OMIdl → RngElt`](ideal-op.md#function-norm-omidl) - [`Valuation(alpha, P : parameters): FldArithElt, OMIdl → RngIntElt, FldElt`](ideal-op.md#function-valuation-fldarithelt-omidl) - [`Valuation(alpha, P : parameters): FldRatElt, OMIdl → RngIntElt, FldElt`](ideal-op.md#function-valuation-fldratelt-omidl) - [`Valuation(alpha, P : parameters): RngIntElt, OMIdl → RngIntElt, FldElt`](ideal-op.md#function-valuation-rngintelt-omidl) - [`Valuation(alpha, P : parameters): RngUPolElt, OMIdl → RngIntElt, FldElt`](ideal-op.md#function-valuation-rngupolelt-omidl) - [`Valuation(I, P): OMIdl, OMIdl → RngIntElt`](ideal-op.md#function-valuation-omidl-omidl) - [`a mod P: FldArithElt, OMIdl → FldArithElt`](ideal-op.md#operation-operation-mod-fldarithelt-omidl-fldarithelt) - [`Reduction(a, P): FldArithElt, OMIdl → FldArithElt`](ideal-op.md#function-reduction-fldarithelt-omidl) - [`Reduction(a, P, m): FldArithElt, OMIdl, RngIntElt → [FldArithElt]`](ideal-op.md#function-reduction-fldarithelt-omidl-rngintelt) - [`Factorization(I): OMIdl → SeqEnum`](ideal-op.md#function-factorization-omidl) - [`Factorisation(I): OMIdl → SeqEnum`](ideal-op.md#function-factorisation-omidl) - [`Example: Om Ideal Ops`](ideal-op.md#example-ex-b5c67b) - [`ResidueField(I): OMIdl → Fld`](ideal-op.md#function-residuefield-omidl) - [`Example: Om Ideals Deg Res`](ideal-op.md#example-ex-7694e8)