Algebraically Closed Fields
- Introduction
- Representation
- Creation of Structures
- Creation of Elements
- Coercion
- Roots
Roots(f): RngUPolElt → [ < FldACElt, RngIntElt> ]Roots(f, A): RngUPolElt, FldAC → [ < FldACElt, RngIntElt> ]RootOfUnity(n, A): RngIntElt, FldAC → FldACEltSquareRoot(a): FldACElt → FldACEltSqrt(a): FldACElt → FldACEltIsSquare(a): FldACElt → BoolEltRoot(a, n): FldACElt, RngIntElt → FldACEltIsPower(a, n): FldACElt, RngIntElt → BoolElt, FldACElt
- Variables
- Related Structures
- Properties
- Ring Predicates and Properties
IsCommutative(A): FldAC → BoolEltIsUnitary(A): FldAC → BoolEltIsFinite(A): FldAC → BoolEltIsOrdered(A): FldAC → BoolEltIsField(A): FldAC → BoolEltIsEuclideanDomain(A): FldAC → BoolEltIsPID(A): FldAC → BoolEltIsUFD(A): FldAC → BoolEltIsDivisionRing(A): FldAC → BoolEltIsEuclideanRing(A): FldAC → BoolEltIsPrincipalIdealRing(A): FldAC → BoolEltIsDomain(A): FldAC → BoolEltA eq B: FldAC, Rng → BoolEltA ne B: FldAC, Rng → BoolEltCharacteristic(A): FldAC → FldACElt
- Element Operations
- Arithmetic Operators
+ a: FldACElt → FldACElt- a: FldACElt → FldACElta + b: FldACElt, FldACElt → FldACElta - b: FldACElt, FldACElt → FldACElta * b: FldACElt, FldACElt → FldACElta / b: FldACElt, FldACElt → FldACElta ^ k: FldACElt, RngIntElt → FldACElta +:= b: FldACElt, FldACElt → FldACElta -:= b: FldACElt, FldACElt → FldACElta *:= b: FldACElt, FldACElt → FldACElt
- Equality and Membership
- Parent and Category
- Predicates on Ring Elements
IsZero(a): FldACElt → BoolEltIsOne(a): FldACElt → BoolEltIsMinusOne(a): FldACElt → BoolElta eq b: FldACElt, FldACElt → BoolElta ne b: FldACElt, FldACElt → BoolElta in A: FldACElt, Rng → BoolElta notin A: FldACElt, Rng → BoolEltIsNilpotent(a): FldACElt → BoolEltIsIdempotent(a): FldACElt → BoolEltIsUnit(a): FldACElt → BoolEltIsZeroDivisor(a): FldACElt → BoolEltIsRegular(a): FldAC → BoolEltIsIrreducible(a): FldACElt → BoolEltIsPrime(a): FldACElt → BoolElt
- Minimal Polynomial, Norm, and Trace
- Arithmetic Operators
- Simplification
- Absolute Field
AbsoluteAffineAlgebra(A): FldAC → RngUPolResAbsoluteQuotientRing(A): FldAC → RngUPolResAbsolutePolynomial(A): FldACNumberField(A): FldAC → FldNum, MapNumberField(a): FldACElt → FldNum, MapAbsolutize(A): FldAC- Example: Field of Definition of the Cyclic-6 Variety
- Example: Splitting Field of a Degree 8 Polynomial
- Example: Splitting Field of a Quartic as a Number Field
- Example: Splitting Field of Degree 720 as a Tower
- Bibliography