NumberFieldSignature

The signature $(r_1, r_2)$ of $\mathbb{Q}(a)$: its real embeddings, and its pairs of complex conjugate ones.

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NumberFieldSignature(a)$(r_1, r_2)$ with $r_1 + 2r_2$ the degree, $r_1$ counted by a Sturm sequence

Typed signatures

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

Domain: Number theory

Details

• By Dirichlet's unit theorem the units of $\mathcal{O}_K$ have rank $r_1 + r_2 - 1$

Examples

Basic #

one real cube root of 2, and a complex pair

live ↗

In$\mathrm{NumberFieldSignature}(\sqrt[3]{2})$Out$\bigl\lbrack1, 1\bigr\rbrack$

all five roots real

live ↗

In$\mathrm{NumberFieldSignature}(\mathrm{PolynomialRoot}(x^5-5x^3+4x-1, 1))$Out$\bigl\lbrack5, 0\bigr\rbrack$

Applications #

live ↗

In$\mathrm{NumberFieldSignature}(\sqrt{2})$Out$\bigl\lbrack2, 0\bigr\rbrack$

live ↗

In$\mathrm{NumberFieldSignature}(i)$Out$\bigl\lbrack0, 1\bigr\rbrack$

live ↗

In$\mathrm{NumberFieldSignature}(\sqrt[3]{3})$Out$\bigl\lbrack1, 1\bigr\rbrack$

About

Examples	5
Checked against	Wolfram Language 5 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