# Power, Laurent and Puiseux Series

- [Introduction](introduction.md)

  - [Kinds of Series](introduction.md#kinds-of-series)

  - [Puiseux Series](introduction.md#puiseux-series)

  - [Representation of Series](introduction.md#representation-of-series)

  - [Precision](introduction.md#precision)

  - [Free and Fixed Precision](introduction.md#free-and-fixed-precision)

  - [Equality](introduction.md#equality)

  - [Polynomials over Series Rings](introduction.md#polynomials-over-series-rings)

- [Creation Functions](creation.md)

  - [Creation of Structures](creation.md#creation-of-structures)

    - [`PowerSeriesRing(R): Rng → RngSerPow`](creation.md#function-powerseriesring-rng)

    - [`PowerSeriesRing(R, p): Rng, RngIntElt → RngSerPow`](creation.md#function-powerseriesring-rng-rngintelt)

    - [`LaurentSeriesRing(R): Rng → RngSerLaur`](creation.md#function-laurentseriesring-rng)

    - [`LaurentSeriesRing(R, p): Rng, RngIntElt → RngSerLaur`](creation.md#function-laurentseriesring-rng-rngintelt)

    - [`PuiseuxSeriesRing(R): Rng → RngSerPuis`](creation.md#function-puiseuxseriesring-rng)

    - [`PuiseuxSeriesRing(R, p): Rng, RngIntElt → RngSerPuis`](creation.md#function-puiseuxseriesring-rng-rngintelt)

    - [`Example: Creation`](creation.md#example-ex-f4cf13)

  - [Special Options](creation.md#special-options)

    - [`AssertAttribute(S, "DefaultPrecision", n): RngSer, MonStgElt, RngIntElt`](creation.md#function-assertattribute-rngser-monstgelt-rngintelt)

    - [`HasAttribute(S, "DefaultPrecision"): RngSer, MonStgElt → BoolElt, RngIntElt`](creation.md#function-hasattribute-rngser-monstgelt)

    - [`AssignNames(~S, ["x"]): RngSer, [ MonStgElt ]`](creation.md#function-assignnames-rngser-monstgelt-ref)

    - [`Name(S, 1): RngSer, RngIntElt → RngSerElt`](creation.md#function-name-rngser-rngintelt)

    - [`S . 1: RngSer, RngIntElt → RngSerElt`](creation.md#operation-operation-rngser-rngintelt-rngserelt)

    - [`Name(S, 1): RngSer, RngIntElt → RngSerElt`](creation.md#function-name-rngser-rngintelt-2)

    - [`S . 1: RngSer, RngIntElt → RngSerElt`](creation.md#operation-operation-rngser-rngintelt-rngserelt-2)

  - [Creation of Elements](creation.md#creation-of-elements)

    - [`R . 1: RngSer, RngInt → RngSerElt`](creation.md#operation-operation-rngser-rngint-rngserelt)

    - [`UniformizingElement(R): RngSer → RngSerElt`](creation.md#function-uniformizingelement-rngser)

    - [`elt< R | v, [ a₁, ..., a_d], p >: RngIntElt, SeqEnum, RngIntElt → RngSerElt`](creation.md#constructor-constructor-elt-rngintelt-seqenum-rngintelt-rngserelt)

    - [`R ! s: RngSer, SeqEnum → RngSerElt`](creation.md#operation-op-rngser-seqenum)

    - [`BigO(f): RngSerElt → RngIntElt`](creation.md#function-bigo-rngserelt)

    - [`O(f): RngSerElt → RngIntElt`](creation.md#function-o-rngserelt)

    - [`One(Q): RngSer → RngSerElt`](creation.md#function-one-rngser)

    - [`One(Q): RngSer → RngSerElt`](creation.md#function-one-rngser-2)

    - [`Identity(Q): RngSer → RngSerElt`](creation.md#function-identity-rngser)

    - [`Identity(Q): RngSer → RngSerElt`](creation.md#function-identity-rngser-2)

    - [`Zero(Q): RngSer → RngSerElt`](creation.md#function-zero-rngser)

    - [`Zero(Q): RngSer → RngSerElt`](creation.md#function-zero-rngser-2)

    - [`Representative(Q): RngSer → RngSerElt`](creation.md#function-representative-rngser)

    - [`Representative(Q): RngSer → RngSerElt`](creation.md#function-representative-rngser-2)

- [Structure Operations](operation-structure.md)

  - [Related Structures](operation-structure.md#related-structures)

    - [`Parent(R): RngSer → Pow`](operation-structure.md#function-parent-rngser)

    - [`Category(R): RngSer → Cat`](operation-structure.md#function-category-rngser)

    - [`BaseRing(R): RngSer → Rng`](operation-structure.md#function-basering-rngser)

    - [`CoefficientRing(R): RngSer → Rng`](operation-structure.md#function-coefficientring-rngser)

    - [`IntegerRing(R): RngSer → RngSerPow`](operation-structure.md#function-integerring-rngser)

    - [`Integers(R): RngSer → RngSerPow`](operation-structure.md#function-integers-rngser)

    - [`RingOfIntegers(R): RngSer → RngSerPow`](operation-structure.md#function-ringofintegers-rngser)

    - [`FieldOfFractions(R): RngSer → RngSerLaur`](operation-structure.md#function-fieldoffractions-rngser)

    - [`ChangePrecision(R, r): RngSer, Any → RngSer`](operation-structure.md#function-changeprecision-rngser-any)

    - [`ChangePrecision(~R, r): RngSer, Any → RngSer`](operation-structure.md#function-changeprecision-rngser-any-ref)

    - [`ChangeRing(R, C): RngSer, Rng → RngSer, Map`](operation-structure.md#function-changering-rngser-rng)

    - [`ResidueClassField(R): RngSer → Rng, Map`](operation-structure.md#function-residueclassfield-rngser)

  - [Invariants](operation-structure.md#invariants)

    - [`Characteristic(R): RngSer → RngIntElt`](operation-structure.md#function-characteristic-rngser)

    - [`Precision(R): RngSer → ExtReElt`](operation-structure.md#function-precision-rngser)

    - [`GetPrecision(R): RngSer → ExtReElt`](operation-structure.md#function-getprecision-rngser)

  - [Ring Predicates and Booleans](operation-structure.md#ring-predicates-and-booleans)

    - [`IsCommutative(Q): RngSer → BoolElt`](operation-structure.md#function-iscommutative-rngser)

    - [`IsUnitary(Q): RngSer → BoolElt`](operation-structure.md#function-isunitary-rngser)

    - [`IsFinite(Q): RngSer → BoolElt`](operation-structure.md#function-isfinite-rngser)

    - [`IsOrdered(Q): RngSer → BoolElt`](operation-structure.md#function-isordered-rngser)

    - [`IsField(Q): RngSer → BoolElt`](operation-structure.md#function-isfield-rngser)

    - [`IsEuclideanDomain(Q): RngSer → BoolElt`](operation-structure.md#function-iseuclideandomain-rngser)

    - [`IsPID(Q): RngSer → BoolElt`](operation-structure.md#function-ispid-rngser)

    - [`IsUFD(Q): RngSer → BoolElt`](operation-structure.md#function-isufd-rngser)

    - [`IsDivisionRing(Q): RngSer → BoolElt`](operation-structure.md#function-isdivisionring-rngser)

    - [`IsEuclideanRing(Q): RngSer → BoolElt`](operation-structure.md#function-iseuclideanring-rngser)

    - [`IsPrincipalIdealRing(Q): RngSer → BoolElt`](operation-structure.md#function-isprincipalidealring-rngser)

    - [`IsDomain(Q): RngSer → BoolElt`](operation-structure.md#function-isdomain-rngser)

    - [`R eq S: RngSer, RngSer → BoolElt`](operation-structure.md#operation-op-eq-rngser-rngser)

    - [`R ne S: RngSer, RngSer → BoolElt`](operation-structure.md#operation-op-ne-rngser-rngser)

- [Basic Element Operations](operation-element.md)

  - [Parent and Category](operation-element.md#parent-and-category)

    - [`Parent(r): RngSerElt → RngSer`](operation-element.md#function-parent-rngserelt)

    - [`Category(r): RngSerElt → Cat`](operation-element.md#function-category-rngserelt)

  - [Arithmetic Operators](operation-element.md#arithmetic-operators)

    - [`+ b: RngSerElt → RngSerElt`](operation-element.md#operation-operation-rngserelt-rngserelt)

    - [`- b: RngSerElt → RngSerElt`](operation-element.md#operation-operation-rngserelt-rngserelt-2)

    - [`a + b: RngSerElt, RngSerElt → RngSerElt`](operation-element.md#operation-op-plus-rngserelt-rngserelt)

    - [`a - b: RngSerElt, RngSerElt → RngSerElt`](operation-element.md#operation-op-minus-rngserelt-rngserelt)

    - [`a * b: RngSerElt, RngSerElt → RngSerElt`](operation-element.md#operation-op-times-rngserelt-rngserelt)

    - [`a ^ k: RngSerElt, RngIntElt → RngSerElt`](operation-element.md#operation-op-pow-rngserelt-rngintelt)

    - [`a div b: RngSerPowElt, RngSerPowElt → RngSerPowElt`](operation-element.md#operation-operation-div-rngserpowelt-rngserpowelt-rngserpowelt)

    - [`a / b: RngSerElt, RngSerElt → RngSerElt`](operation-element.md#operation-op-div-rngserelt-rngserelt)

  - [Equality and Membership](operation-element.md#equality-and-membership)

    - [`a eq b: RngSerElt, RngSerElt → BoolElt`](operation-element.md#operation-op-eq-rngserelt-rngserelt)

    - [`a ne b: RngSerElt, RngSerElt → BoolElt`](operation-element.md#operation-op-ne-rngserelt-rngserelt)

    - [`a in R: RngSerElt, RngSer → BoolElt`](operation-element.md#operation-op-in-rngserelt-rngser)

    - [`a notin R: RngSerElt, RngSer → BoolElt`](operation-element.md#operation-op-notin-rngserelt-rngser)

  - [Predicates on Ring Elements](operation-element.md#predicates-on-ring-elements)

    - [`IsZero(a): RngSerElt → BoolElt`](operation-element.md#function-iszero-rngserelt)

    - [`IsOne(a): RngSerElt → BoolElt`](operation-element.md#function-isone-rngserelt)

    - [`IsMinusOne(a): RngSerElt → BoolElt`](operation-element.md#function-isminusone-rngserelt)

    - [`IsNilpotent(x): RngSerElt → BoolElt`](operation-element.md#function-isnilpotent-rngserelt)

    - [`IsIdempotent(x): RngSerElt → BoolElt`](operation-element.md#function-isidempotent-rngserelt)

    - [`IsUnit(a): RngSerElt → BoolElt`](operation-element.md#function-isunit-rngserelt)

    - [`IsZeroDivisor(x): RngSerElt → BoolElt`](operation-element.md#function-iszerodivisor-rngserelt)

    - [`IsRegular(x): RngSerElt → BoolElt`](operation-element.md#function-isregular-rngserelt)

    - [`IsIrreducible(x): RngSerElt → BoolElt`](operation-element.md#function-isirreducible-rngserelt)

    - [`IsPrime(x): RngSerElt → BoolElt`](operation-element.md#function-isprime-rngserelt)

    - [`IsWeaklyZero(f): RngSerElt → BoolElt`](operation-element.md#function-isweaklyzero-rngserelt)

    - [`IsWeaklyEqual(f, g): RngSerElt, RngSerElt → BoolElt`](operation-element.md#function-isweaklyequal-rngserelt-rngserelt)

    - [`IsIdentical(f, g): RngSerElt, RngSerElt → BoolElt`](operation-element.md#function-isidentical-rngserelt-rngserelt)

  - [Precision](operation-element.md#precision)

    - [`AbsolutePrecision(f): RngSerElt → RngIntElt`](operation-element.md#function-absoluteprecision-rngserelt)

    - [`RelativePrecision(f): RngSerElt → RngIntElt`](operation-element.md#function-relativeprecision-rngserelt)

    - [`ChangePrecision(f, r): RngSerElt, RngIntElt → RngSerElt`](operation-element.md#function-changeprecision-rngserelt-rngintelt)

    - [`ChangePrecision(f, r): RngSerElt, Infty → RngSerElt`](operation-element.md#function-changeprecision-rngserelt-infty)

    - [`ChangePrecision(~f, r): RngSerElt, RngIntElt`](operation-element.md#function-changeprecision-rngserelt-rngintelt-ref)

    - [`ChangePrecision(~f, r): RngSerElt, Infty`](operation-element.md#function-changeprecision-rngserelt-infty-ref)

  - [Coefficients and Degree](operation-element.md#coefficients-and-degree)

    - [`Coefficients(f): RngSerElt → [ RngElt ], RngIntElt, RngIntElt`](operation-element.md#function-coefficients-rngserelt)

    - [`ElementToSequence(f): RngSerElt → [ RngElt ], RngIntElt, RngIntElt`](operation-element.md#function-elementtosequence-rngserelt)

    - [`Eltseq(f): RngSerElt → [ RngElt ], RngIntElt, RngIntElt`](operation-element.md#function-eltseq-rngserelt)

    - [`Coefficient(f, i): RngSerElt, RngElt → RngElt`](operation-element.md#function-coefficient-rngserelt-rngelt)

    - [`LeadingCoefficient(f): RngSerElt → RngElt`](operation-element.md#function-leadingcoefficient-rngserelt)

    - [`LeadingTerm(f): RngSerElt → RngElt`](operation-element.md#function-leadingterm-rngserelt)

    - [`Truncate(f): RngSerElt → RngSerElt`](operation-element.md#function-truncate-rngserelt)

    - [`ExponentDenominator(f): RngMSerElt → RngElt`](operation-element.md#function-exponentdenominator-rngmserelt)

    - [`Degree(f): RngSerElt → RngIntElt`](operation-element.md#function-degree-rngserelt)

    - [`Valuation(f): RngSerElt → RngIntElt`](operation-element.md#function-valuation-rngserelt)

    - [`ExponentDenominator(f): RngSerElt → RngIntElt`](operation-element.md#function-exponentdenominator-rngserelt)

  - [Evaluation and Derivative](operation-element.md#evaluation-and-derivative)

    - [`Derivative(f): RngSerElt → RngSerElt`](operation-element.md#function-derivative-rngserelt)

    - [`Derivative(f, n): RngSerElt, RngIntElt → RngSerElt`](operation-element.md#function-derivative-rngserelt-rngintelt)

    - [`Integral(f): RngSerElt → RngSerElt`](operation-element.md#function-integral-rngserelt)

    - [`Evaluate(f, s): RngSerElt, RngElt → RngElt`](operation-element.md#function-evaluate-rngserelt-rngelt)

    - [`Laplace(f): RngSerElt → RngSerElt`](operation-element.md#function-laplace-rngserelt)

  - [Square Root](operation-element.md#square-root)

    - [`SquareRoot(f): RngSerElt → RngSerElt`](operation-element.md#function-squareroot-rngserelt)

    - [`Sqrt(f): RngSerElt → RngSerElt`](operation-element.md#function-sqrt-rngserelt)

  - [Composition and Reversion](operation-element.md#composition-and-reversion)

    - [`Composition(f, g): RngSerElt, RngSerElt → RngSerElt`](operation-element.md#function-composition-rngserelt-rngserelt)

    - [`Reversion(f): RngSerElt → RngSerElt`](operation-element.md#function-reversion-rngserelt)

    - [`Reverse(f): RngSerElt → RngSerElt`](operation-element.md#function-reverse-rngserelt)

    - [`Convolution(f, g): RngSerElt, RngSerElt → RngSerElt`](operation-element.md#function-convolution-rngserelt-rngserelt)

    - [`Example: Composition Reversion`](operation-element.md#example-ex-cad173)

- [Transcendental Functions](transcendental.md)

  - [Exponential and Logarithmic Functions](transcendental.md#exponential-and-logarithmic-functions)

    - [`Exp(f): RngSerElt → RngSerElt`](transcendental.md#function-exp-rngserelt)

    - [`Log(f): RngSerElt → RngSerElt`](transcendental.md#function-log-rngserelt)

    - [`Example: Bernoulli`](transcendental.md#example-ex-739f49)

  - [Trigonometric Functions and their Inverses](transcendental.md#trigonometric-functions-and-their-inverses)

    - [`Sin(f): RngSerElt → RngSerElt`](transcendental.md#function-sin-rngserelt)

    - [`Cos(f): RngSerElt → RngSerElt`](transcendental.md#function-cos-rngserelt)

    - [`Sincos(f): RngSerElt → RngSerElt`](transcendental.md#function-sincos-rngserelt)

    - [`Tan(f): RngSerElt → RngSerElt`](transcendental.md#function-tan-rngserelt)

    - [`Arcsin(f): RngSerElt → RngSerElt`](transcendental.md#function-arcsin-rngserelt)

    - [`Arccos(f): RngSerElt → RngSerElt`](transcendental.md#function-arccos-rngserelt)

    - [`Arctan(f): RngSerElt → RngSerElt`](transcendental.md#function-arctan-rngserelt)

  - [Hyperbolic Functions and their Inverses](transcendental.md#hyperbolic-functions-and-their-inverses)

    - [`Sinh(f): RngSerElt → RngSerElt`](transcendental.md#function-sinh-rngserelt)

    - [`Cosh(f): RngSerElt → RngSerElt`](transcendental.md#function-cosh-rngserelt)

    - [`Tanh(f): RngSerElt → RngSerElt`](transcendental.md#function-tanh-rngserelt)

    - [`Argsinh(f): RngSerElt → RngSerElt`](transcendental.md#function-argsinh-rngserelt)

    - [`Argcosh(f): RngSerElt → RngSerElt`](transcendental.md#function-argcosh-rngserelt)

    - [`Argtanh(f): RngSerElt → RngSerElt`](transcendental.md#function-argtanh-rngserelt)

- [The Hypergeometric Series](hypergeometric.md)

  - [`HypergeometricSeries(a,b,c, z): RngElt, RngElt, RngElt, RngElt → RngElt`](hypergeometric.md#function-hypergeometricseries-rngelt-rngelt-rngelt-rngelt)

- [Polynomials over Series Rings](polynomials.md)

  - [`HenselLift(f, L): RngUPolElt[RngSer], SeqEnum[RngUPolElt] → [RngUPolElt]`](polynomials.md#function-hensellift-rngupolelt-rngser-seqenum-rngupolelt)

  - [`HenselLift(f, L): RngUPolElt[RngSerExt], SeqEnum[RngUPolElt] → [RngUPolElt]`](polynomials.md#function-hensellift-rngupolelt-rngserext-seqenum-rngupolelt)

  - [`Factorization(f): RngUPolElt[RngSerPow[FldFin]] → [ < RngUPolElt[RngSerPow], RngIntElt > ], RngSerPowElt`](polynomials.md#function-factorization-rngupolelt-rngserpow-fldfin)

  - [`Factorization(f): RngUPolElt[RngSerLaur[FldFin]] → [ < RngUPolElt[RngSerLaur], RngIntElt > ], RngSerLaurElt`](polynomials.md#function-factorization-rngupolelt-rngserlaur-fldfin)

  - [`Factorization(f): RngUPolElt[RngSerExt] → [ < RngUPolElt[RngSerExt], RngIntElt > ], RngSerExtElt`](polynomials.md#function-factorization-rngupolelt-rngserext)

  - [`Example: Series Poly Fact`](polynomials.md#example-ex-32baab)

- [Extensions of Series Rings](extensions.md)

  - [Constructions of Extensions](extensions.md#constructions-of-extensions)

    - [`UnramifiedExtension(R, f): RngSerPow[FldFin], RngUPolElt → RngSerExt`](extensions.md#function-unramifiedextension-rngserpow-fldfin-rngupolelt)

    - [`UnramifiedExtension(R, f): RngSerLaur[FldFin], RngUPolElt → RngSerExt`](extensions.md#function-unramifiedextension-rngserlaur-fldfin-rngupolelt)

    - [`UnramifiedExtension(R, f): RngSerExt, RngUPolElt → RngSerExt`](extensions.md#function-unramifiedextension-rngserext-rngupolelt)

    - [`TotallyRamifiedExtension(R, f): RngSerPow[FldFin], RngUPolElt → RngSerExt`](extensions.md#function-totallyramifiedextension-rngserpow-fldfin-rngupolelt)

    - [`TotallyRamifiedExtension(R, f): RngSerLaur[FldFin], RngUPolElt → RngSerExt`](extensions.md#function-totallyramifiedextension-rngserlaur-fldfin-rngupolelt)

    - [`TotallyRamifiedExtension(R, f): RngSerExt, RngUPolElt → RngSerExt`](extensions.md#function-totallyramifiedextension-rngserext-rngupolelt)

    - [`ChangePrecision(E, r): RngSerExt, RngIntElt → RngSerExt`](extensions.md#function-changeprecision-rngserext-rngintelt)

    - [`ChangePrecision(~E, r): RngSerExt, RngIntElt`](extensions.md#function-changeprecision-rngserext-rngintelt-ref)

    - [`FieldOfFractions(E): RngSerExt → RngSerExt`](extensions.md#function-fieldoffractions-rngserext)

    - [`Example: Extensions Eg`](extensions.md#example-ex-0fa010)

  - [Operations on Extensions](extensions.md#operations-on-extensions)

    - [`Precision(E): RngSerExt → RngIntElt`](extensions.md#function-precision-rngserext)

    - [`GetPrecision(E): RngSerExt → RngIntElt`](extensions.md#function-getprecision-rngserext)

    - [`CoefficientRing(E): RngSerExt → Rng`](extensions.md#function-coefficientring-rngserext)

    - [`BaseRing(E): RngSerExt → Rng`](extensions.md#function-basering-rngserext)

    - [`DefiningPolynomial(E): RngSerExt → RngUPolElt`](extensions.md#function-definingpolynomial-rngserext)

    - [`InertiaDegree(E): RngSerExt → RngIntElt`](extensions.md#function-inertiadegree-rngserext)

    - [`RamificationIndex(E): RngSerExt → RngIntElt`](extensions.md#function-ramificationindex-rngserext)

    - [`RamificationDegree(E): RngSerExt → RngIntElt`](extensions.md#function-ramificationdegree-rngserext)

    - [`ResidueClassField(E): RngSerExt → FldFin`](extensions.md#function-residueclassfield-rngserext)

    - [`UniformizingElement(E): RngSerExt → RngSerExtElt`](extensions.md#function-uniformizingelement-rngserext)

    - [`IntegerRing(E): RngSerExt → RngSerExt`](extensions.md#function-integerring-rngserext)

    - [`Integers(E): RngSerExt → RngSerExt`](extensions.md#function-integers-rngserext)

    - [`RingOfIntegers(E): RngSerExt → RngSerExt`](extensions.md#function-ringofintegers-rngserext)

    - [`E1 eq E2: RngSerExt, RngSerExt → BoolElt`](extensions.md#operation-op-eq-rngserext-rngserext)

    - [`E . i: RngSerExt, RngIntElt → RngSerExtElt`](extensions.md#operation-operation-rngserext-rngintelt-rngserextelt)

    - [`AssignNames(~E, S): RngSerExt, [ MonStgElt ]`](extensions.md#function-assignnames-rngserext-monstgelt-ref)

    - [`Example: Ext Ops`](extensions.md#example-ex-b88c0a)

  - [Elements of Extensions](extensions.md#elements-of-extensions)

    - [`x * y: RngSerExtElt, RngSerExtElt → RngSerExtElt`](extensions.md#operation-op-times-rngserextelt-rngserextelt)

    - [`x + y: RngSerExtElt, RngSerExtElt → RngSerExtElt`](extensions.md#operation-op-plus-rngserextelt-rngserextelt)

    - [`x - y: RngSerExtElt, RngSerExtElt → RngSerExtElt`](extensions.md#operation-op-minus-rngserextelt-rngserextelt)

    - [`- x: RngSerExtElt → RngSerExtElt`](extensions.md#operation-operation-rngserextelt-rngserextelt)

    - [`x ^ n: RngSerExtElt, RngIntElt → RngSerExtElt`](extensions.md#operation-op-pow-rngserextelt-rngintelt)

    - [`x div y: RngSerExtElt, RngSerExtElt → RngSerExtElt`](extensions.md#operation-operation-div-rngserextelt-rngserextelt-rngserextelt)

    - [`x / y: RngSerExtElt, RngSerExtElt → RngSerExtElt`](extensions.md#operation-op-div-rngserextelt-rngserextelt)

    - [`x eq y: RngSerExtElt, RngSerExtElt → BoolElt`](extensions.md#operation-op-eq-rngserextelt-rngserextelt)

    - [`IsZero(e): RngSerExtElt → BoolElt`](extensions.md#function-iszero-rngserextelt)

    - [`IsOne(e): RngSerExtElt → BoolElt`](extensions.md#function-isone-rngserextelt)

    - [`IsMinusOne(e): RngSerExtElt → BoolElt`](extensions.md#function-isminusone-rngserextelt)

    - [`IsUnit(e): RngSerExtElt → BoolElt`](extensions.md#function-isunit-rngserextelt)

    - [`Valuation(e): RngSerExtElt → RngIntElt`](extensions.md#function-valuation-rngserextelt)

    - [`RelativePrecision(e): RngSerExtElt → RngIntElt`](extensions.md#function-relativeprecision-rngserextelt)

    - [`AbsolutePrecision(e): RngSerExtElt → RngIntElt`](extensions.md#function-absoluteprecision-rngserextelt)

    - [`Coefficients(e): RngSerExtElt → [ RngElt ]`](extensions.md#function-coefficients-rngserextelt)

    - [`Eltseq(e): RngSerExtElt → [ RngElt ]`](extensions.md#function-eltseq-rngserextelt)

    - [`ElementToSequence(e): RngSerExtElt → [ RngElt ]`](extensions.md#function-elementtosequence-rngserextelt)

    - [`Example: Serext Simple`](extensions.md#example-ex-77b403)

  - [Optimized Representation](extensions.md#optimized-representation)

    - [`OptimizedRepresentation(E): RngSerExt → RngSer, Map`](extensions.md#function-optimizedrepresentation-rngserext)

    - [`OptimisedRepresentation(E): RngSerExt → RngSer, Map`](extensions.md#function-optimisedrepresentation-rngserext)

    - [`Example: Opt Rep`](extensions.md#example-ex-95694d)