# 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](../IntroductionToRings/index-introduction-to-rings.md#rngintro)) 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`

```magma
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`

```magma
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 ]`

```magma
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: Element Ops (ex-c52d75)`

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

```magma
> 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!