AlgebraicOrder

The order $\mathbb{Z}[a]$ generated by the algebraic integer $a$: the polynomials in $a$ over $\mathbb{Z}$, a subring of finite index in the ring of integers of $\mathbb{Q}(a)$.

SageMathNumberField(f, 'a').order(a)↗

AlgebraicOrder(a)$\mathbb{Z}[a]$, for $a$ an algebraic integer; in degree 2 it is the QuadraticOrder of its discriminant (or QuadraticIntegers(d) when it is the whole ring)

Typed signatures

Type	Package	Applies
(number) -> set<number>	number-theory	always

Domain: Number theory

Details

• Named by a generator because past degree 2 orders aren't determined by their discriminant; its index in AlgebraicIntegers is $\sqrt{\mathrm{disc}\,\mathbb{Z}[a] / d_K}$
• A value for Over, as AlgebraicIntegers is; at a prime dividing the index, $\mathcal{O}/(x)$ being a field is still what makes $(x)$ prime

Examples

Basic #

$\mathbb{Z}[\sqrt{-3}]$, of conductor 2

live ↗

In$\mathrm{AlgebraicOrder}(\sqrt{-3})$Out$\mathrm{QuadraticOrder}(-12)$

$\mathbb{Z}[\sqrt[3]{2}]$ is all of $\mathcal{O}_K$, but keeps its name

live ↗

In$\mathrm{AlgebraicOrder}(\sqrt[3]{2})$Out$\mathrm{AlgebraicOrder}(\sqrt[3]{2})$

3 does not divide the index 2, so it is prime in $\mathbb{Z}[\theta]$ as in $\mathcal{O}_K$

live ↗

In$\mathrm{IsPrime}(3, \operatorname{KeyValuePair}(\mathrm{Over}, \mathrm{AlgebraicOrder}(\mathrm{PolynomialRoot}(x^3-x^2-2x-8, 1))))$Out$\mathrm{True}$

About

Examples	3
Crosswalk	SageMath

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram has no orders besides the maximal one.

See also: AlgebraicIntegers, QuadraticOrder, MinimalPolynomial