QuadraticInteger

A name from the enumeratio catalogue with no kernel of its own yet; this page carries its crosswalk only.

The element $x + y\omega$ of QuadraticIntegers(d), held as a value.

QuadraticInteger(d, x, y)$x + y\omega$, with $\omega = (-1 + \sqrt d)/2$ when $d \equiv 1 \pmod 4$, else $\sqrt d$

Typed signatures

Type	Package	Applies
(integer | tuple<integer, integer, integer>, integer?, integer?) -> quadratic_integer	number-theory	always

Domain: Number theory

Details

• The carrier names its own ring, so a head reads it without Over: IsPrime(QuadraticInteger(-5, 1, 1)) asks about $1 + \sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$
• $\omega$ is the ring's own generator, so $x, y$ are always integers: QuadraticInteger(-3, 0, 1) is $\omega = (-1 + \sqrt{-3})/2 = e^{2\pi i/3}$
• QuadraticInteger(-1, x, y) is GaussianInteger(x, y)

Examples

Basic #

1 + √−5 is irreducible but not prime

live ↗

In$\mathrm{IsPrime}(\mathrm{QuadraticInteger}(-5, 1, 1))$Out$\mathrm{False}$

ω = (−1 + √−3)/2 is a unit

live ↗

In$\mathrm{IsPrime}(\mathrm{QuadraticInteger}(-3, 0, 1))$Out$\mathrm{False}$

2 splits in ℤ[ω], ω = (−1 + √−7)/2

live ↗

In$\mathrm{FactorInteger}(\mathrm{QuadraticInteger}(-7, 4, 0))$Out$\bigl\lbrack(\frac{-7^{1/2}}{2}-\frac{1}{2},2), (\frac{1}{2}+\frac{\sqrt{-7}}{2},2)\bigr\rbrack$

Scope #

QuadraticInteger(-1, x, y) is x + yi

live ↗

In$\mathrm{IsPrime}(\mathrm{QuadraticInteger}(-1, 2, 1))$Out$\mathrm{True}$

About

Examples	4

See also: QuadraticIntegers, GaussianInteger