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ALGEBRAICALLY CLOSED FIELDS

 
Acknowledgements
 
Introduction
 
Representation
 
Creation of Structures
 
Creation of Elements
      Coercion
      Roots
      Variables
 
Related Structures
 
Properties
 
Ring Predicates and Properties
 
Element Operations
      Arithmetic Operators
      Equality and Membership
      Parent and Category
      Predicates on Ring Elements
      Minimal Polynomial, Norm and Trace
 
Simplification
 
Absolute Field
 
Bibliography







DETAILS

 
Introduction

 
Representation

 
Creation of Structures
      AlgebraicClosure(K) : Fld -> FldAC
      AlgebraicClosure() : -> FldAC
      AssignNamePrefix(A, S) : FldAC, MonStgElt ->

 
Creation of Elements

      Coercion
            A ! a : FldAC, RngElt -> FldACElt

      Roots
            Roots(f) : RngUPolElt -> [ < FldACElt, RngIntElt> ]
            RootOfUnity(n, A) : RngIntElt, FldAC -> FldACElt
            SquareRoot(a) : FldACElt -> FldACElt
            IsSquare(a) : FldACElt -> BoolElt
            Root(a, n) : FldACElt, RngIntElt -> FldACElt
            IsPower(a, n) : FldACElt, RngIntElt -> BoolElt, FldACElt

      Variables
            A . i : FldAC, RngIntElt -> FldACElt
            Example FldAC_Create (H40E1)
            Example FldAC_SwinnertonDyer (H40E2)
            Example FldAC_Puiseux (H40E3)

 
Related Structures

 
Properties
      BaseField(A) : FldAC -> Fld
      Rank(A) : FldAC -> RngIntElt
      Degree(A, v) : FldAC, RngIntElt -> RngIntElt
      Degree(A) : FldAC -> RngIntElt
      AffineAlgebra(A) : FldAC -> RngMPolRes
      Ideal(A) : FldAC -> RngMPol

 
Ring Predicates and Properties

 
Element Operations

      Arithmetic Operators

      Equality and Membership

      Parent and Category

      Predicates on Ring Elements
            IsZero(a) : FldACElt -> BoolElt
            IsOne(a) : FldACElt -> BoolElt
            IsMinusOne(a) : FldACElt -> BoolElt
            a eq b : FldACElt, FldACElt -> BoolElt

      Minimal Polynomial, Norm and Trace
            MinimalPolynomial(a) : FldACElt -> RngUPolElt
            Norm(a) : FldACElt -> FldACElt
            Trace(a) : FldACElt -> FldACElt
            Conjugates(a) : FldACElt -> [ FldACElt ]
            Example FldAC_Functions (H40E4)

 
Simplification
      Simplify(A) : FldAC ->
      Prune(A) : FldAC ->

 
Absolute Field
      AbsoluteAffineAlgebra(A) : FldAC -> RngUPolRes
      AbsolutePolynomial(A) : FldAC ->
      Absolutize(A) : FldAC ->
      Example FldAC_Cyclic6 (H40E5)
      Example FldAC_Split (H40E6)

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013