AlgebraicIntegers

The ring of integers $\mathcal{O}_K$ of the number field $K = \mathbb{Q}(a)$: every element of $K$ that is a root of a monic polynomial over $\mathbb{Z}$, its maximal order.

SageMathNumberField(f, 'a').maximal_order()↗

AlgebraicIntegers(a)the maximal order of $\mathbb{Q}(a)$, for $a$ a radical of a rational, $i$, the golden ratio or a PolynomialRoot; in degree 1 it is Integers, in degree 2 QuadraticIntegers(d)

Typed signatures

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

Domain: Number theory

Details

• A value for Over: IsPrime(x, Over -> AlgebraicIntegers(a)) asks whether $(x)$ is a prime ideal, that is whether $\mathcal{O}_K/(x)$ is a field. With $|N(x)| = p^f$ that is $p \in (x)$, Frobenius injective on the quotient and fixing only $\mathbb{F}_p$, all linear algebra over $\mathbb{F}_p$
• $\mathcal{O}_K$ is found as NumberFieldDiscriminant finds it, and stays unevaluated where that declines
• Past degree 2 it stays AlgebraicIntegers(a); NumberFieldIntegralBasis lists a basis

Examples

Basic #

$\mathbb{Z}[i]$, already named

live ↗

In$\mathrm{AlgebraicIntegers}(i)$Out$\mathrm{GaussianIntegers}$

$\mathbb{Z}[\varphi]$ is the ring of integers of $\mathbb{Q}(\sqrt 5)$

live ↗

In$\mathrm{AlgebraicIntegers}(\varphi)$Out$\mathrm{QuadraticIntegers}(5)$

$x^3 - x^2 - 2x - 8$ is irreducible mod 3, so 3 stays prime

live ↗

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

2 splits into three primes of norm 2, which is why no $\mathbb{Z}[\alpha]$ is all of $\mathcal{O}_K$

live ↗

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

its norm is 3, a prime: $(1 + \sqrt[3]{2})$ is a prime ideal of degree 1

live ↗

In$\mathrm{IsPrime}(1+\sqrt[3]{2}, \operatorname{KeyValuePair}(\mathrm{Over}, \mathrm{AlgebraicIntegers}(\sqrt[3]{2})))$Out$\mathrm{True}$

About

Examples	5
Crosswalk	SageMath

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram names no ring of integers: its NumberField* functions take the algebraic number and work in this ring.

See also: AlgebraicOrder, QuadraticIntegers, NumberFieldIntegralBasis, IsPrime