NumberFieldIntegralBasis

A $\mathbb{Z}$-basis of the ring of integers of $\mathbb{Q}(a)$, written on $a$'s powers as Wolfram writes it.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageNumberFieldIntegralBasis✓↗

NumberFieldIntegralBasis(a)a basis of $\mathcal{O}_K$: the $i$-th element $a^i/d_i$ plus lower powers, each lower coefficient reduced

Typed signatures

Type	Package	Applies
(number) -> list	number-theory	always

Domain: Number theory

Details

• The basis is triangular on $1, a, \ldots, a^{n-1}$, as Wolfram's is; any other basis differs by a matrix in $\mathrm{GL}_n(\mathbb{Z})$

Examples

Basic #

Dedekind's: $1, \theta, (\theta + \theta^2)/2$

live ↗

In$\mathrm{NumberFieldIntegralBasis}(\mathrm{PolynomialRoot}(x^3-x^2-2x-8, 1))$Out$\bigl\lbrack1, \mathrm{PolynomialRoot}(x^3-x^2-2x-8, 1), \frac{1}{2}(\mathrm{PolynomialRoot}(x^3-x^2-2x-8, 1))+\frac{1}{2}({\mathrm{PolynomialRoot}(x^3-x^2-2x-8, 1)}^2)\bigr\rbrack$

$1, \theta, (1 + \theta + \theta^2)/3$

live ↗

In$\mathrm{NumberFieldIntegralBasis}(\sqrt[3]{19})$Out$\bigl\lbrack1, \sqrt[3]{19}, \frac{1}{3}+\frac{\sqrt[3]{19}}{3}+\frac{19^{\frac{2}{3}}}{3}\bigr\rbrack$

$1, \omega$

live ↗

In$\mathrm{NumberFieldIntegralBasis}(\sqrt{-3})$Out$\bigl\lbrack1, \frac{1}{2}+\frac{\sqrt{-3}}{2}\bigr\rbrack$

About

Examples	3
Checked against	Wolfram Language 3 agree
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/ce-patches/src/compute-engine/library/algebraic-numbers.ts

mappedwolframpackages/oracle/src/mappings.ts

See also: NumberFieldDiscriminant, AlgebraicIntegers