Subsections

• Arithmetic Operators
• Equality and Membership
• Parent and Category
• Predicates on Ring Elements
• Other Element Functions
• Solving Linear Equations in Z/mZ

+ n : RngIntResElt -> RngIntResElt

- n : RngIntResElt -> RngIntResElt

m + n : RngIntResElt, RngIntResElt -> RngIntResElt

m - n : RngIntResElt, RngIntResElt -> RngIntResElt

m * n : RngIntResElt, RngIntResElt -> RngIntResElt

n ^ k : RngIntResElt, RngIntResElt -> RngIntResElt

m / n : RngIntResElt, RngIntResElt -> RngIntResElt

m +:= n : RngIntResElt, RngIntResElt -> RngIntResElt

m -:= n : RngIntResElt, RngIntResElt -> RngIntResElt

m *:= n : RngIntResElt, RngIntResElt -> RngIntResElt

m /:= n : RngIntResElt, RngIntResElt -> RngIntResElt

m ^:= k : RngIntResElt, RngIntResElt -> RngIntResElt

m eq n : RngIntResElt, RngIntResElt -> BoolElt

m ne n : RngInRngIntRest, RngIntResElt -> BoolElt

n in R : RngIntResElt, Rng -> BoolElt

n notin R : RngIntResElt, Rng -> BoolElt

Parent(n) : RngIntResElt -> RngIntRes

Category(n) : RngIntResElt -> Cat

IsZero(n) : RngIntResElt -> BoolElt

IsOne(n) : RngIntResElt -> BoolElt

IsMinusOne(n) : RngIntResElt -> BoolElt

IsNilpotent(n) : RngIntResElt -> BoolElt

IsIdempotent(n) : RngIntResElt -> BoolElt

IsUnit(n) : RngIntResElt -> BoolElt

IsZeroDivisor(n) : RngIntResElt -> BoolElt

IsRegular(n) : RngIntRes -> BoolElt

IsIrreducible(n) : RngIntResElt -> BoolElt

IsPrime(n) : RngIntResElt -> BoolElt

    Factorization: [ < int, int > ]     Default: [ ]

Given an element n inZ/mZ this function returns true if there exists a inZ/mZ such that a^2=n inZ/mZ, false otherwise. If n is a square, a square root a is also returned. If m is large and its prime factorization is known, the computation may be speeded up by assigning the factorization sequence for m to the optional argument Factorization.

Returns true if the element n inZ/mZ is primitive, that is, if it generates the multiplicative group of Z/mZ, false otherwise.

PrimitiveRoot(R) : RngIntRes -> RngIntResElt

Given R = Z/mZ, this function returns a generator for the group of units of R if this group is cyclic, and returns 0 otherwise. (Thus a valid generator is only returned if m = 2, 4, p^t or 2p^t, with p an odd prime and t >= 1.)

Given an element a belonging to Z/mZ, return the multiplicative order k >= 1 of a if a is in the unit group (Z/mZ)^*, and zero if a is not a unit.

SquareRoot(a) : RngIntResElt -> RngIntResElt

    Factorization: [ < int, int > ]     Default: [ ]

Given an element a of the ring Z/mZ, this function returns an element b of Z/mZ such that b^2=a inZ/mZ, if such an element exists, and an error otherwise. If m is large and its prime factorization is known, the computation may be speeded up by assigning the factorization sequence for m to the optional argument Factorization.

AllSqrts(a) : RngIntResElt -> [ RngIntResElt ]

    Factorization: [ < int, int > ]     Default: [ ]

Return a sequence of all square roots of an element a in a residue class ring Z/mZ. If the modulus m is large and its prime factorization is known, the computation may be speeded up by assigning the factorization sequence for m to the optional argument Factorization.

Given elements a and b of Z/mZ, return a solution x to the linear congruence a.x=b in Z/mZ. An error is signalled if no solution exists.