AlgebraicNumberNorm

The norm of $a$ over $\mathbb{Q}$ in the field $\mathbb{Q}(a)$: the product of its conjugates.

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

AlgebraicNumberNorm(a)the product of $a$'s conjugates, $(-1)^d$ times its minimal polynomial's constant term over its leading one

Typed signatures

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

Domain: Number theory

Details

• In $\mathbb{Q}(a)$, the field $a$ generates; in a larger field of degree $n$ the norm is this to the power $n/d$

Examples

Basic #

$(1 + \sqrt 2)(1 - \sqrt 2) = -1$: a unit

live ↗

In$\mathrm{AlgebraicNumberNorm}(1+\sqrt{2})$Out$-1$

the three cube roots of 2 multiply to 2

live ↗

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

in $\mathbb{Q}$ itself

live ↗

In$\mathrm{AlgebraicNumberNorm}(3)$Out$3$

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: AlgebraicNumberTrace, MinimalPolynomial