QuadraticIntegers

The ring of integers of $\mathbb{Q}(\sqrt d)$: $\mathbb{Z}[\omega]$ with $\omega = (-1 + \sqrt d)/2$ when $d \equiv 1 \pmod 4$, else $\sqrt d$.

SageMathQuadraticField(d).ring_of_integers()↗mathlib4NumberField.RingOfIntegers↗

QuadraticIntegers(d)the ring of integers of $\mathbb{Q}(\sqrt d)$, for a non-square integer $d$; a square factor of $d$ names the same field

SageMathQuadraticField(d).ring_of_integers()

Typed signatures

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

Domain: Number theory

Details

• A value for the Over option: IsPrime(x, Over -> QuadraticIntegers(-5)) asks whether $x$ is prime in $\mathbb{Z}[\sqrt{-5}]$. IsPrime, IsComposite, FactorInteger, Divisors, PrimeNu, PrimeOmega, MoebiusMu and IsSquareFree read it
• QuadraticIntegers(-1) is GaussianIntegers, answered with the Gaussian conventions
• Primality holds in every ring: $x$ is prime when the ideal $(x)$ is prime. Most of these rings do not factor uniquely ($6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ in $\mathbb{Z}[\sqrt{-5}]$), so FactorInteger, Divisors and the counting heads answer only where the class number is 1, and stay unevaluated elsewhere
• An irreducible that is not prime is not composite either: IsComposite(2, Over -> QuadraticIntegers(-5)) is False, and so is IsPrime
• The guide Quadratic integers draws these rings as lattices

Examples

Basic #

2 is irreducible in ℤ[√−5] but not prime: it divides (1 + √−5)(1 − √−5) without dividing either

live ↗

In$\mathrm{IsPrime}(2, \operatorname{KeyValuePair}(\mathrm{Over}, \mathrm{QuadraticIntegers}(-5)))$Out$\mathrm{False}$

nor composite: it has no proper factorization

live ↗

In$\mathrm{IsComposite}(2, \operatorname{KeyValuePair}(\mathrm{Over}, \mathrm{QuadraticIntegers}(-5)))$Out$\mathrm{False}$

live ↗

In$\mathrm{IsPrime}(1+\sqrt{-5}, \operatorname{KeyValuePair}(\mathrm{Over}, \mathrm{QuadraticIntegers}(-5)))$Out$\mathrm{False}$

6 = 2 · 3 = (1 + √−5)(1 − √−5): no prime factorization to give

live ↗

In$\mathrm{FactorInteger}(6, \operatorname{KeyValuePair}(\mathrm{Over}, \mathrm{QuadraticIntegers}(-5)))$Out$\mathrm{FactorInteger}(6, (\mathrm{Over},\mathrm{QuadraticIntegers}(-5)))$

7 = (1 + 2√2)(−1 + 2√2) in ℤ[√2]

live ↗

In$\mathrm{FactorInteger}(7, \operatorname{KeyValuePair}(\mathrm{Over}, \mathrm{QuadraticIntegers}(2)))$Out$\bigl\lbrack(1+2\sqrt{2},1), (2\sqrt{2}-1,1)\bigr\rbrack$

Scope #

7 splits in the Eisenstein integers; the unit comes first

live ↗

In$\mathrm{FactorInteger}(7, \operatorname{KeyValuePair}(\mathrm{Over}, \mathrm{QuadraticIntegers}(-3)))$Out$\bigl\lbrack(\frac{1}{2}-\frac{\sqrt{-3}}{2},1), (2+\sqrt{-3},1), (\frac{5}{2}+\frac{\sqrt{-3}}{2},1)\bigr\rbrack$

QuadraticIntegers(-1) is GaussianIntegers

live ↗

In$\mathrm{FactorInteger}(5, \operatorname{KeyValuePair}(\mathrm{Over}, \mathrm{QuadraticIntegers}(-1)))$Out$\bigl\lbrack(-i,1), (1+2i,1), (2+i,1)\bigr\rbrack$

About

Examples	7
Crosswalk	SageMath ×2mathlib4

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram has no ring-of-integers value to pass as a domain: its number-theory heads take GaussianIntegers -> True for Z[i] only, and NumberFieldClassNumber and friends take an algebraic number, not a ring.

mappedsagepackages/oracle/src/mappings.ts

See also: QuadraticInteger, GaussianIntegers, IsPrime, FactorInteger