build open-axiom
10574
(|AlgebraicallyClosedField|)
|category|
(((|AlgebraicallyClosedField|) (|Category|)) (T |AlgebraicallyClosedField|))
(T)
(|Join| (|Field|) (|RadicalCategory|)
(CATEGORY |domain| (SIGNATURE |rootOf| ($ (|Polynomial| $)))
(SIGNATURE |rootOf| ($ (|SparseUnivariatePolynomial| $)))
(SIGNATURE |rootOf| ($ (|SparseUnivariatePolynomial| $) (|Symbol|)))
(SIGNATURE |rootsOf| ((|List| $) (|Polynomial| $)))
(SIGNATURE |rootsOf| ((|List| $) (|SparseUnivariatePolynomial| $)))
(SIGNATURE |rootsOf|
((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)))
(SIGNATURE |zeroOf| ($ (|Polynomial| $)))
(SIGNATURE |zeroOf| ($ (|SparseUnivariatePolynomial| $)))
(SIGNATURE |zeroOf| ($ (|SparseUnivariatePolynomial| $) (|Symbol|)))
(SIGNATURE |zerosOf| ((|List| $) (|Polynomial| $)))
(SIGNATURE |zerosOf| ((|List| $) (|SparseUnivariatePolynomial| $)))
(SIGNATURE |zerosOf|
((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)))))
"algfunc.spad"
((|rootOf| (*1 *1 *2)
(AND (|isDomain| *2 (|Polynomial| *1))
(|ofCategory| *1 (|AlgebraicallyClosedField|))))
(|rootOf| (*1 *1 *2)
(AND (|isDomain| *2 (|SparseUnivariatePolynomial| *1))
(|ofCategory| *1 (|AlgebraicallyClosedField|))))
(|rootOf| (*1 *1 *2 *3)
(AND (|isDomain| *2 (|SparseUnivariatePolynomial| *1))
(|isDomain| *3 (|Symbol|))
(|ofCategory| *1 (|AlgebraicallyClosedField|))))
(|rootsOf| (*1 *2 *3)
(AND (|isDomain| *3 (|Polynomial| *1))
(|ofCategory| *1 (|AlgebraicallyClosedField|))
(|isDomain| *2 (|List| *1))))
(|rootsOf| (*1 *2 *3)
(AND (|isDomain| *3 (|SparseUnivariatePolynomial| *1))
(|ofCategory| *1 (|AlgebraicallyClosedField|))
(|isDomain| *2 (|List| *1))))
(|rootsOf| (*1 *2 *3 *4)
(AND (|isDomain| *3 (|SparseUnivariatePolynomial| *1))
(|isDomain| *4 (|Symbol|))
(|ofCategory| *1 (|AlgebraicallyClosedField|))
(|isDomain| *2 (|List| *1))))
(|zeroOf| (*1 *1 *2)
(AND (|isDomain| *2 (|Polynomial| *1))
(|ofCategory| *1 (|AlgebraicallyClosedField|))))
(|zeroOf| (*1 *1 *2)
(AND (|isDomain| *2 (|SparseUnivariatePolynomial| *1))
(|ofCategory| *1 (|AlgebraicallyClosedField|))))
(|zeroOf| (*1 *1 *2 *3)
(AND (|isDomain| *2 (|SparseUnivariatePolynomial| *1))
(|isDomain| *3 (|Symbol|))
(|ofCategory| *1 (|AlgebraicallyClosedField|))))
(|zerosOf| (*1 *2 *3)
(AND (|isDomain| *3 (|Polynomial| *1))
(|ofCategory| *1 (|AlgebraicallyClosedField|))
(|isDomain| *2 (|List| *1))))
(|zerosOf| (*1 *2 *3)
(AND (|isDomain| *3 (|SparseUnivariatePolynomial| *1))
(|ofCategory| *1 (|AlgebraicallyClosedField|))
(|isDomain| *2 (|List| *1))))
(|zerosOf| (*1 *2 *3 *4)
(AND (|isDomain| *3 (|SparseUnivariatePolynomial| *1))
(|isDomain| *4 (|Symbol|))
(|ofCategory| *1 (|AlgebraicallyClosedField|))
(|isDomain| *2 (|List| *1)))))
((~= (#1=((|Boolean|) $ $) 7 T ELT))
(|zerosOf| (((|List| $) (|Polynomial| $)) 99 T ELT)
(((|List| $) (|SparseUnivariatePolynomial| $)) 98 T ELT)
(((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)) 97 T ELT))
(|zeroOf| (($ (|Polynomial| $)) 102 T ELT)
(($ (|SparseUnivariatePolynomial| $)) 101 T ELT)
(($ (|SparseUnivariatePolynomial| $) (|Symbol|)) 100 T ELT))
(|zero?| ((#2=(|Boolean|) $) 22 T ELT))
(|unitNormal|
(((|Record| (|:| |unit| $) (|:| |canonical| $) (|:| |associate| $)) $) 56 T
ELT))
(|unitCanonical| (($ $) 55 T ELT)) (|unit?| ((#3=(|Boolean|) $) 53 T ELT))
(|subtractIfCan| (((|Maybe| $) $ $) 26 T ELT))
(|squareFreePart| (($ $) 92 T ELT))
(|squareFree| (#4=((|Factored| $) $) 91 T ELT)) (|sqrt| (($ $) 111 T ELT))
(|sizeLess?| (((|Boolean|) $ $) 76 T ELT)) (|sample| (#5=($) 23 T CONST))
(|rootsOf| (((|List| $) (|Polynomial| $)) 105 T ELT)
(((|List| $) (|SparseUnivariatePolynomial| $)) 104 T ELT)
(((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)) 103 T ELT))
(|rootOf| (($ (|Polynomial| $)) 108 T ELT)
(($ (|SparseUnivariatePolynomial| $)) 107 T ELT)
(($ (|SparseUnivariatePolynomial| $) (|Symbol|)) 106 T ELT))
(|rem| (#6=($ $ $) 72 T ELT)) (|recip| (((|Union| $ "failed") $) 43 T ELT))
(|quo| (#6# 73 T ELT))
(|principalIdeal|
(((|Record| (|:| |coef| #7=(|List| $)) (|:| |generator| $)) #7#) 67 T ELT))
(|prime?| (((|Boolean|) $) 90 T ELT)) (|opposite?| ((#2# $ $) 20 T ELT))
(|one?| (((|Boolean|) $) 45 T ELT))
(|nthRoot| (($ $ #8=(|Integer|)) 110 T ELT))
(|multiEuclidean| (((|Union| #9=(|List| $) #10="failed") #9# $) 69 T ELT))
(|lcm| (#11=($ $ $) 61 T ELT) (#12=($ (|List| $)) 60 T ELT))
(|latex| (((|String|) $) 11 T ELT)) (|inv| (($ $) 89 T ELT))
(|hash| (((|SingleInteger|) $) 12 T ELT))
(|gcdPolynomial| ((#13=(|SparseUnivariatePolynomial| $) #13# #13#) 59 T ELT))
(|gcd| (#11# 63 T ELT) (#12# 62 T ELT)) (|factor| (#4# 93 T ELT))
(|extendedEuclidean|
(((|Record| #14=(|:| |coef1| $) #15=(|:| |coef2| $) (|:| |generator| $)) $ $)
71 T ELT)
(((|Union| (|Record| #14# #15#) #10#) $ $ $) 70 T ELT))
(|exquo| (((|Union| $ "failed") $ $) 57 T ELT))
(|expressIdealMember| (((|Maybe| #7#) #7# $) 66 T ELT))
(|euclideanSize| (((|NonNegativeInteger|) $) 75 T ELT))
(|divide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) 74 T ELT))
(|coerce| (((|OutputForm|) $) 13 T ELT) (($ (|Integer|)) 42 T ELT)
(($ $) 58 T ELT) (($ #16=(|Fraction| #17=(|Integer|))) 85 T ELT))
(|characteristic| (((|NonNegativeInteger|)) 41 T CONST))
(|before?| (#1# 6 T ELT)) (|associates?| ((#3# $ $) 54 T ELT))
(|annihilate?| (((|Boolean|) $ $) 34 T ELT)) (|Zero| (#5# 24 T CONST))
(|One| (($) 46 T CONST)) (= (#1# 8 T ELT)) (/ (($ $ $) 84 T ELT))
(- (($ $) 30 T ELT) (($ $ $) 29 T ELT)) (+ (($ $ $) 18 T ELT))
(** (($ $ (|PositiveInteger|)) 36 T ELT)
(($ $ (|NonNegativeInteger|)) 44 T ELT) (($ $ #17#) 88 T ELT)
(($ $ (|Fraction| #8#)) 109 T ELT))
(* (($ (|PositiveInteger|) $) 17 T ELT)
(($ (|NonNegativeInteger|) $) 21 T ELT)
(($ (|Integer|) . #18=($)) 31 T ELT) (($ $ $) 35 T ELT)
(($ $ #16#) 87 T ELT) (($ #16# . #18#) 86 T ELT)))
((|canonicalUnitNormal| . T) (|noZeroDivisors| . T) ((|commutative| "*") . T))
ACF
(((|Field|) . T) ((|RadicalCategory|) . T))
(((|AbelianGroup|) . T) ((|AbelianMonoid|) . T) ((|AbelianSemiGroup|) . T)
((|Algebra| #1=(|Fraction| (|Integer|))) . T) ((|Algebra| $) . T)
((|BasicType|) . T) ((|BiModule| #1# #1#) . T) ((|BiModule| $ $) . T)
((|CancellationAbelianMonoid|) . T) ((|CoercibleFrom| #1#) . T)
((|CoercibleFrom| $) . T) ((|CoercibleFrom| (|Integer|)) . T)
((|CoercibleTo| (|OutputForm|)) . T) ((|CommutativeRing|) . T)
((|DivisionRing|) . T) ((|EntireRing|) . T) ((|EuclideanDomain|) . T)
((|Field|) . T) ((|GcdDomain|) . T) ((|IntegralDomain|) . T)
((|LeftLinearSet| #1#) . T) ((|LeftLinearSet| $) . T)
((|LeftLinearSet| (|Integer|)) . T) ((|LeftModule| #1#) . T)
((|LeftModule| $) . T) ((|LinearSet| #1#) . T) ((|LinearSet| $) . T)
((|Module| #1#) . T) ((|Module| $) . T) ((|Monoid|) . T)
((|PrincipalIdealDomain|) . T) ((|RadicalCategory|) . T)
((|RightLinearSet| #1#) . T) ((|RightLinearSet| $) . T)
((|RightModule| #1#) . T) ((|RightModule| $) . T) ((|Ring|) . T) ((|Rng|) . T)
((|SemiGroup|) . T) ((|SemiRing|) . T) ((|SetCategory|) . T) ((|Type|) . T)
((|UniqueFactorizationDomain|) . T))
((|constructor| (NIL "Model for algebraically closed fields."))
(|zerosOf|
(((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|))
"\\spad{zerosOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}'s are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities which display as \\spad{'yi}. The returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.")
(((|List| $) (|SparseUnivariatePolynomial| $))
"\\spad{zerosOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}'s are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities. The returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.")
(((|List| $) (|Polynomial| $))
"\\spad{zerosOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}'s are expressed in radicals if possible. Otherwise they are implicit algebraic quantities. The returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}."))
(|zeroOf|
(($ (|SparseUnivariatePolynomial| $) (|Symbol|))
"\\spad{zeroOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}; if possible,{} \\spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity which displays as \\spad{'y}.")
(($ (|SparseUnivariatePolynomial| $))
"\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}; if possible,{} \\spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity.")
(($ (|Polynomial| $))
"\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. If possible,{} \\spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity. Error: if \\spad{p} has more than one variable \\spad{y}."))
(|rootsOf|
(((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|))
"\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}. Note: the returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.")
(((|List| $) (|SparseUnivariatePolynomial| $))
"\\spad{rootsOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. Note: the returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.")
(((|List| $) (|Polynomial| $))
"\\spad{rootsOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. Note: the returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}."))
(|rootOf|
(($ (|SparseUnivariatePolynomial| $) (|Symbol|))
"\\spad{rootOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}.")
(($ (|SparseUnivariatePolynomial| $))
"\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}.")
(($ (|Polynomial| $))
"\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. Error: if \\spad{p} has more than one variable \\spad{y}.")))
(("documentation" 0 7370) ("ancestors" 0 6277) ("parents" 0 6233)
("abbreviation" 0 6229) ("predicates" 0 NIL) ("attributes" 0 6150)
("signaturesAndLocals" 0 NIL) ("superDomain" 0 NIL) ("operationAlist" 0 3000)
("modemaps" 0 1079) ("sourceFile" 0 1064) ("constructorCategory" 0 142)
("dualSignature" 0 138) ("constructorModemap" 0 61) ("constructorKind" 0 50)
("constructorForm" 0 21))