Elements of Residue Class Rings

Creation

elt< R | k >: RngIntRes, RngIntElt -> RngIntResElt

Create the residue class containing the integer \(k\) in residue class ring \(R\).

R ! k: RngIntRes, RngIntElt -> RngIntResElt

Create the residue class containing \(k\) in the residue class ring \(R\). Here \(k\) is allowed to be either an integer, or an element of the finite field \({\mathbb{F}}_p\) in the case \(R = {\mathbb{Z}}/p{\mathbb{Z}}\), or an element of \(S = {\mathbb{Z}}/n{\mathbb{Z}}\) for a multiple or divisor \(n\) of \(m\) (with \(R = {{\mathbb{Z}}/m{\mathbb{Z}}}\)).

One(R): RngIntRes -> RngIntResElt
Identity(R): RngIntRes -> RngIntResElt
Zero(R): RngIntRes -> RngIntResElt
Representative(R): RngIntRes -> RngIntResElt

These generic functions (cf. Chapter Introduction to Rings) create \(1\), \(1\), \(0\), and \(0\) respectively, in any \({{\mathbb{Z}}/m{\mathbb{Z}}}\).

Random(R): RngIntRes -> RngIntResElt

Create a “random” residue class in \(R\).

Arithmetic Operators

+ 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 div 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

Equality and Membership

m eq n: RngIntResElt, RngIntResElt -> BoolElt
m ne n: RngIntResElt, RngIntResElt -> BoolElt
n in R: RngIntResElt, Rng -> BoolElt
n notin R: RngIntResElt, Rng -> BoolElt

Parent and Category

Parent(n): RngIntResElt -> RngIntRes
Category(n): RngIntResElt -> Cat

Predicates on Ring Elements

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

Solving Equations over \({\mathbb{Z}}/m{\mathbb{Z}}\)

Solution(a, b): RngIntResElt, RngIntResElt -> RngIntResElt

Given elements \(a\) and \(b\) of \({\mathbb{Z}}/m{\mathbb{Z}}\), return a solution \(x\) to the linear congruence \(a\cdot x=b \in {{\mathbb{Z}}/m{\mathbb{Z}}}\). An error is signalled if no solution exists.

IsSquare(n): RngIntResElt -> BoolElt, RngIntResElt
Factorization: [<RngIntElt, RngIntElt>]                    Default: [ ]

Given an element \(n\in{{\mathbb{Z}}/m{\mathbb{Z}}}\) this function returns true if there exists \(a\in{{\mathbb{Z}}/m{\mathbb{Z}}}\) such that \(a^2=n\in{{\mathbb{Z}}/m{\mathbb{Z}}}\), 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.

Sqrt(a): RngIntResElt -> RngIntResElt
SquareRoot(a): RngIntResElt -> RngIntResElt
Factorization: [<RngIntElt, RngIntElt>]                    Default: [ ]

Given an element \(a\) of the ring \({\mathbb{Z}}/m{\mathbb{Z}}\), this function returns an element \(b\) of \({\mathbb{Z}}/m{\mathbb{Z}}\) such that \(b^2=a\in{{\mathbb{Z}}/m{\mathbb{Z}}}\), 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.

AllSquareRoots(a): RngIntResElt -> [ RngIntResElt ]
AllSqrts(a): RngIntResElt -> [ RngIntResElt ]
Factorization: [<RngIntElt, RngIntElt>]                    Default: [ ]

Return a sequence containing all square roots of the element \(a\) in a residue class ring \({\mathbb{Z}}/m{\mathbb{Z}}\). 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.

Example: Square Roots Modulo 2340 (ex-c52d75)

We construct the residue class ring having modulus \(2340\) and find all the square roots of \(1404\).

> R := ResidueClassRing(2340);
Residue class ring of integers modulo 2340
> x := R!1404;
> sqrts := AllSquareRoots(x);
> sqrts;
[ 78, 312, 468, 702, 858, 1092, 1248, 1482, 1638,
  1872, 2028, 2262 ]
> [ y^2 : y in sqrts ];
[ 1404, 1404, 1404, 1404, 1404, 1404, 1404, 1404,
  1404, 1404, 1404, 1404 ]

So 1404 has 12 square roots!

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