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NumberFieldRegulator[a]

gives the regulator of the field generated by the algebraic number a.

Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
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NumberFieldRegulator

NumberFieldRegulator[a]

gives the regulator of the field generated by the algebraic number a.

Examples

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Basic Examples  (1)

Find the regulator of the field :

Wolfram Language code: NumberFieldRegulator[Sqrt[2] + I]

Scope  (4)

Radical objects:

Wolfram Language code: NumberFieldRegulator[Sqrt[2] + Sqrt[3]]

Root objects:

Wolfram Language code: NumberFieldRegulator[Root[2 + 3#1 ^ 3&, 2]]

AlgebraicNumber objects:

Wolfram Language code: NumberFieldRegulator[AlgebraicNumber[Root[-10 + #1^3&, 1], {1, 0, 1 / 3}]]

NumberFieldRegulator automatically threads over lists:

Wolfram Language code: NumberFieldRegulator[{Sqrt[2], E ^ (Pi * I / 4)}]

Applications  (2)

Find the volume of the fundamental domain of the lattice generated by fundamental units:

Wolfram Language code: NumberFieldRegulator[#]Sqrt[Length[NumberFieldFundamentalUnits[#]] + 1]&[Sqrt[2] + Sqrt[3]]
Wolfram Language code: N[%]

The Dirichlet L-functions at 1 for the real primitive character Mod13:

Wolfram Language code: 2NumberFieldClassNumber[Sqrt[13]]NumberFieldRegulator[Sqrt[13]] / Sqrt[13]

Properties & Relations  (2)

The regulator of an imaginary quadratic field is 1:

Wolfram Language code: NumberFieldRegulator[Sqrt[-2]]

Use ToNumberField to find the regulator of the field :

Wolfram Language code: primitive = ToNumberField[{Sqrt[2], Sqrt[-5]}, All][[1, 1]]
Wolfram Language code: NumberFieldRegulator[primitive]
Wolfram Research (2007), NumberFieldRegulator, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldRegulator.html.

Text

Wolfram Research (2007), NumberFieldRegulator, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldRegulator.html.

CMS

Wolfram Language. 2007. "NumberFieldRegulator." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NumberFieldRegulator.html.

APA

Wolfram Language. (2007). NumberFieldRegulator. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NumberFieldRegulator.html

BibTeX

@misc{reference.wolfram_2026_numberfieldregulator, author="Wolfram Research", title="{NumberFieldRegulator}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/NumberFieldRegulator.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_numberfieldregulator, organization={Wolfram Research}, title={NumberFieldRegulator}, year={2007}, url={https://reference.wolfram.com/language/ref/NumberFieldRegulator.html}, note=[Accessed: 12-June-2026]}