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

As we explained in the previous section, Julia types don't contain enough information to properly model the ring of integers modulo $n$ for a multiprecision modulus $n$. Instead of using types to construct objects, 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.

Mathematics	Nemo 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")
$S = R[x]$ (to precision $n$)	S, x = PowerSeriesRing(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")
$O = \mathcal{O}_K$	S = MaximalOrder(K)
ideal $I$ of $O = \mathcal{O}_K$	I = Ideal(O, gens, ...)