# Algebraically Closed Fields - [Introduction](introduction.md) - [Representation](representation.md) - [Creation of Structures](structure.md) - [`AlgebraicClosure(K): Fld → FldAC`](structure.md#function-fldac-algebraicclosure) - [`AlgebraicClosure() → FldAC`](structure.md#function-algebraicclosure) - [`AssignNamePrefix(A, S): FldAC, MonStgElt`](structure.md#function-assignnameprefix-fldac-monstgelt) - [Creation of Elements](creation.md) - [Coercion](creation.md#coercion) - [`A ! a: FldAC, RngElt → FldACElt`](creation.md#operation-op-fldac-rngelt) - [`One(A): FldAC → FldACElt`](creation.md#function-one-fldac) - [`Identity(A): FldAC → FldACElt`](creation.md#function-identity-fldac) - [`Zero(A): FldAC → FldACElt`](creation.md#function-zero-fldac) - [`Representative(A): FldAC → FldACElt`](creation.md#function-representative-fldac) - [Roots](creation.md#roots) - [`Roots(f): RngUPolElt → [ < FldACElt, RngIntElt> ]`](creation.md#function-fldac-roots) - [`Roots(f, A): RngUPolElt, FldAC → [ < FldACElt, RngIntElt> ]`](creation.md#function-roots-rngupolelt-fldac) - [`RootOfUnity(n, A): RngIntElt, FldAC → FldACElt`](creation.md#function-rootofunity-rngintelt-fldac) - [`SquareRoot(a): FldACElt → FldACElt`](creation.md#function-squareroot-fldacelt) - [`Sqrt(a): FldACElt → FldACElt`](creation.md#function-fldac-sqrt) - [`IsSquare(a): FldACElt → BoolElt`](creation.md#function-issquare-fldacelt) - [`Root(a, n): FldACElt, RngIntElt → FldACElt`](creation.md#function-root-fldacelt-rngintelt) - [`IsPower(a, n): FldACElt, RngIntElt → BoolElt, FldACElt`](creation.md#function-ispower-fldacelt-rngintelt) - [Variables](creation.md#variables) - [`A . i: FldAC, RngIntElt → FldACElt`](creation.md#operation-operation-fldac-rngintelt-fldacelt) - [`Example: Create`](creation.md#example-ex-f1defe) - [`Example: Swinnerton Dyer`](creation.md#example-ex-681d9c) - [`Example: Puiseux`](creation.md#example-ex-efef60) - [Related Structures](related.md) - [`Category(A): FldAC → Cat`](related.md#function-category-fldac) - [`Parent(A): FldAC → PowerStructure`](related.md#function-parent-fldac) - [`Centre(A): FldAC → AC`](related.md#function-centre-fldac) - [`PrimeRing(A): FldAC → AC`](related.md#function-primering-fldac) - [`PrimeField(A): FldAC → AC`](related.md#function-primefield-fldac) - [`FieldOfFractions(A): FldAC → AC`](related.md#function-fieldoffractions-fldac) - [Properties](property.md) - [`BaseField(A): FldAC → Fld`](property.md#function-basefield-fldac) - [`Rank(A): FldAC → RngIntElt`](property.md#function-rank-fldac) - [`Degree(A, v): FldAC, RngIntElt → RngIntElt`](property.md#function-degree-fldac-rngintelt) - [`Degree(A): FldAC → RngIntElt`](property.md#function-degree-fldac) - [`AffineAlgebra(A): FldAC → RngMPolRes`](property.md#function-fldac-affinealgebra) - [`QuotientRing(A): FldAC → RngMPolRes`](property.md#function-quotientring-fldac) - [`Ideal(A): FldAC → RngMPol`](property.md#function-ideal-fldac) - [Ring Predicates and Properties](predicate.md) - [`IsCommutative(A): FldAC → BoolElt`](predicate.md#function-iscommutative-fldac) - [`IsUnitary(A): FldAC → BoolElt`](predicate.md#function-isunitary-fldac) - [`IsFinite(A): FldAC → BoolElt`](predicate.md#function-isfinite-fldac) - [`IsOrdered(A): FldAC → BoolElt`](predicate.md#function-isordered-fldac) - [`IsField(A): FldAC → BoolElt`](predicate.md#function-isfield-fldac) - [`IsEuclideanDomain(A): FldAC → BoolElt`](predicate.md#function-iseuclideandomain-fldac) - [`IsPID(A): FldAC → BoolElt`](predicate.md#function-ispid-fldac) - [`IsUFD(A): FldAC → BoolElt`](predicate.md#function-isufd-fldac) - [`IsDivisionRing(A): FldAC → BoolElt`](predicate.md#function-isdivisionring-fldac) - [`IsEuclideanRing(A): FldAC → BoolElt`](predicate.md#function-iseuclideanring-fldac) - [`IsPrincipalIdealRing(A): FldAC → BoolElt`](predicate.md#function-isprincipalidealring-fldac) - [`IsDomain(A): FldAC → BoolElt`](predicate.md#function-isdomain-fldac) - [`A eq B: FldAC, Rng → BoolElt`](predicate.md#operation-op-eq-fldac-rng) - [`A ne B: FldAC, Rng → BoolElt`](predicate.md#operation-op-ne-fldac-rng) - [`Characteristic(A): FldAC → FldACElt`](predicate.md#function-characteristic-fldac) - [Element Operations](operation-element.md) - [Arithmetic Operators](operation-element.md#arithmetic-operators) - [`+ a: FldACElt → FldACElt`](operation-element.md#operation-operation-fldacelt-fldacelt) - [`- a: FldACElt → FldACElt`](operation-element.md#operation-operation-fldacelt-fldacelt-2) - [`a + b: FldACElt, FldACElt → FldACElt`](operation-element.md#operation-op-plus-fldacelt-fldacelt) - [`a - b: FldACElt, FldACElt → FldACElt`](operation-element.md#operation-op-minus-fldacelt-fldacelt) - [`a * b: FldACElt, FldACElt → FldACElt`](operation-element.md#operation-op-times-fldacelt-fldacelt) - [`a / b: FldACElt, FldACElt → FldACElt`](operation-element.md#operation-op-div-fldacelt-fldacelt) - [`a ^ k: FldACElt, RngIntElt → FldACElt`](operation-element.md#operation-op-pow-fldacelt-rngintelt) - [`a +:= b: FldACElt, FldACElt → FldACElt`](operation-element.md#operation-operation-fldacelt-fldacelt-fldacelt) - [`a -:= b: FldACElt, FldACElt → FldACElt`](operation-element.md#operation-operation-fldacelt-fldacelt-fldacelt-2) - [`a *:= b: FldACElt, FldACElt → FldACElt`](operation-element.md#operation-operation-fldacelt-fldacelt-fldacelt-3) - [Equality and Membership](operation-element.md#equality-and-membership) - [Parent and Category](operation-element.md#parent-and-category) - [`Parent(a): FldACElt → AC`](operation-element.md#function-parent-fldacelt) - [`Category(a): FldACElt → Cat`](operation-element.md#function-category-fldacelt) - [Predicates on Ring Elements](operation-element.md#predicates-on-ring-elements) - [`IsZero(a): FldACElt → BoolElt`](operation-element.md#function-fldac-iszero) - [`IsOne(a): FldACElt → BoolElt`](operation-element.md#function-isone-fldacelt) - [`IsMinusOne(a): FldACElt → BoolElt`](operation-element.md#function-isminusone-fldacelt) - [`a eq b: FldACElt, FldACElt → BoolElt`](operation-element.md#operation-op-eq-fldacelt-fldacelt) - [`a ne b: FldACElt, FldACElt → BoolElt`](operation-element.md#operation-op-ne-fldacelt-fldacelt) - [`a in A: FldACElt, Rng → BoolElt`](operation-element.md#operation-op-in-fldacelt-rng) - [`a notin A: FldACElt, Rng → BoolElt`](operation-element.md#operation-op-notin-fldacelt-rng) - [`IsNilpotent(a): FldACElt → BoolElt`](operation-element.md#function-isnilpotent-fldacelt) - [`IsIdempotent(a): FldACElt → BoolElt`](operation-element.md#function-isidempotent-fldacelt) - [`IsUnit(a): FldACElt → BoolElt`](operation-element.md#function-isunit-fldacelt) - [`IsZeroDivisor(a): FldACElt → BoolElt`](operation-element.md#function-iszerodivisor-fldacelt) - [`IsRegular(a): FldAC → BoolElt`](operation-element.md#function-isregular-fldac) - [`IsIrreducible(a): FldACElt → BoolElt`](operation-element.md#function-isirreducible-fldacelt) - [`IsPrime(a): FldACElt → BoolElt`](operation-element.md#function-isprime-fldacelt) - [Minimal Polynomial, Norm and Trace](operation-element.md#minimal-polynomial-norm-and-trace) - [`MinimalPolynomial(a): FldACElt → RngUPolElt`](operation-element.md#function-fldac-minimalpolynomial) - [`Norm(a): FldACElt → FldACElt`](operation-element.md#function-norm-fldacelt) - [`Trace(a): FldACElt → FldACElt`](operation-element.md#function-trace-fldacelt) - [`Conjugates(a): FldACElt → [ FldACElt ]`](operation-element.md#function-conjugates-fldacelt) - [`Example: Functions`](operation-element.md#example-ex-e31b1f) - [Simplification](simplify.md) - [`Simplify(A): FldAC`](simplify.md#function-fldac-simplify) - [`Prune(A): FldAC`](simplify.md#function-fldac-prune) - [Absolute Field](absolute.md) - [`AbsoluteAffineAlgebra(A): FldAC → RngUPolRes`](absolute.md#function-absoluteaffinealgebra-fldac) - [`AbsoluteQuotientRing(A): FldAC → RngUPolRes`](absolute.md#function-absolutequotientring-fldac) - [`AbsolutePolynomial(A): FldAC`](absolute.md#function-absolutepolynomial-fldac) - [`Absolutize(A): FldAC`](absolute.md#function-fldac-absolutize) - [`Example: Cyclic6`](absolute.md#example-ex-32dfbd) - [`Example: Split`](absolute.md#example-ex-862d19)