NumberFieldDiscriminant

The discriminant of the number field $\mathbb{Q}(a)$ : of its ring of integers $\mathcal{O}_K$ .

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NumberFieldDiscriminant(a)the discriminant of

\mathcal{O}_K

for

K = \mathbb{Q}(a)

; it divides the minimal polynomial's discriminant by the square of

[\mathcal{O}_K : \mathbb{Z}[a]]

Typed signatures

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

Domain: Number theory

Details

$\mathcal{O}_K$ is found by enlarging $\mathbb{Z}[a]$ at each prime $p$ with $p^2$ dividing its discriminant, while some element of $\frac1p\mathbb{Z}[a]$ outside it is integral; the search declines (stays unevaluated) when it runs past its cap or the discriminant can't be factored

Examples

Basic #

Dedekind's cubic: $x^3 - x^2 - 2x - 8$ has discriminant $-2012 = -4 \cdot 503$ , and $\mathcal{O}_K$ is twice as fine as $\mathbb{Z}[\theta]$

\mathrm{NumberFieldDiscriminant}(\mathrm{PolynomialRoot}(x^3-x^2-2x-8, 1))

Out

-503

$\mathbb{Q}(\sqrt[3]{19})$ : $-3 \cdot 19^2$ , a third of $\mathbb{Z}[\sqrt[3]{19}]$ 's

\mathrm{NumberFieldDiscriminant}(\sqrt[3]{19})

Out

-1\,083

$\mathbb{Q}$

\mathrm{NumberFieldDiscriminant}(3)

Out

1

About

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

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