Constructing mathematical objects in Nemo

# Constructing mathematical objects in Nemo

## Constructing objects in Julia

In Julia, one constructs objects of a given type by calling a type constructor. This is simply a function with the same name as the type itself. For example, to construct a BigInt object in Julia, we simply call the BigInt constructor:

n = BigInt("1234567898765434567898765434567876543456787654567890")

Julia also uses constructors to convert between types. For example, to convert an Int to a BigInt:

m = BigInt(123)

## How we construct objects in Nemo

Julia types don't contain enough information to properly model groups, rings and fields, especially if they are parameterised by values. For example, the ring of integers modulo $n$ for a multiprecision modulus $n$ cannot be modeled using types alone.

Instead of using types to construct objects in Nemo, we use special objects that we refer to as parent objects. They behave a lot like Julia types.

Consider the following simple example, to create a Flint multiprecision integer:

n = ZZ("12345678765456787654567890987654567898765678909876567890")

Here ZZ is not a Julia type, but a callable object. However, for most purposes one can think of such a parent object ZZ as though it were a type.

## Constructing parent objects

For more complicated groups, rings, fields, etc., one first needs to construct the parent object before one can use it to construct element objects.

Nemo provides a set of functions for constructing such parent objects. For example, to create a parent object for polynomials over the integers, we use the PolynomialRing parent object constructor.

R, x = PolynomialRing(ZZ, "x")
f = x^3 + 3x + 1
g = R(12)

In this example, R is the parent object and we use it to convert the Int value $12$ to an element of the polynomial ring $\mathbb{Z}[x]$.

## List of parent object constructors

For convenience, we provide a list of all the parent object constructors in Nemo and explain what domains they represent.

MathematicsNemo constructor
$R = \mathbb{Z}$R = ZZ
$R = \mathbb{Q}$R = QQ
$R = \mathbb{F}_{p^n}$R, a = FiniteField(p, n, "a")
$R = \mathbb{Z}/n\mathbb{Z}$R = ResidueRing(ZZ, n)
$S = R[x]$S, x = PolynomialRing(R, "x")
$S = R[x, y]$S, (x, y, z) = PolynomialRing(R, ["x", "y"])
$S = R[[x]]$ (to precision $n$)S, x = PowerSeriesRing(R, n, "x")
$S = R((x))$ (to precision $n$)S, x = LaurentSeriesRing(R, n, "x")
$S = \mbox{Frac}_R$S = FractionField(R)
$S = R/(f)$S = ResidueRing(R, f)
$S = \mbox{Mat}_{m\times n}(R)$S = MatrixSpace(R, m, n)
$S = \mathbb{Q}[x]/(f)$S, a = NumberField(f, "a")
$S = \mathbb{Q}_p$ (to precision $N$)S = PadicField(p, n)
$S = \mathbb{R}$ (to precision $n$)S = RealField(n)
$S = \mathbb{C}$ (to precision $n$)S = ComplexField(n)`