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AlgebraicNumberNorm[a]

gives the norm of the algebraic number a.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Extension  
Applications  
Properties & Relations  
Neat Examples  
See Also
Tech Notes
Related Guides
History
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AlgebraicNumberNorm

AlgebraicNumberNorm[a]

gives the norm of the algebraic number a.

Details and Options

  • The norm of a is defined to be the product of the roots of its minimal polynomial.
  • AlgebraicNumberNorm[a,Extension->θ] finds the norm of a over the field .

Examples

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Basic Examples  (1)

Norms of algebraic numbers:

Wolfram Language code: AlgebraicNumberNorm[Sqrt[2]]
Wolfram Language code: AlgebraicNumberNorm[E ^ (2Pi I) / 3]
Wolfram Language code: AlgebraicNumberNorm[GoldenRatio]

Scope  (4)

Integers and rational numbers:

Wolfram Language code: AlgebraicNumberNorm[2]
Wolfram Language code: AlgebraicNumberNorm[-2 / 3]

Radical expressions:

Wolfram Language code: AlgebraicNumberNorm[1 / Sqrt[Sqrt[2] + 3]]

Root and AlgebraicNumber objects:

Wolfram Language code: AlgebraicNumberNorm[Root[-1 + #1 + #1 ^ 2 + #1 ^ 3 + #1 ^ 4 & , 1]]
Wolfram Language code: AlgebraicNumberNorm[AlgebraicNumber[Sqrt[2] I, {1, 2}]]

AlgebraicNumberNorm automatically threads over lists:

Wolfram Language code: AlgebraicNumberNorm[{2Sqrt[2], E ^ (Pi * I / 8), 1 + I}]

Options  (1)

Extension  (1)

Norm of over :

Wolfram Language code: AlgebraicNumberNorm[Sqrt[2], Extension -> E ^ (Pi I / 4)]

Applications  (1)

is irreducible in :

Wolfram Language code: AlgebraicNumberNorm[9 + Sqrt[10]]

Since AlgebraicNumberNorm is multiplicative, having a prime norm implies the original number is prime:

Wolfram Language code: PrimeQ[%]

Properties & Relations  (3)

AlgebraicNumberNorm is multiplicative:

Wolfram Language code: AlgebraicNumberNorm[{2, Sqrt[5]}, Extension -> Sqrt[5]]
Wolfram Language code: AlgebraicNumberNorm[2 Sqrt[5], Extension -> Sqrt[5]]

Units in a number field have norm :

Wolfram Language code: AlgebraicNumberNorm /@ NumberFieldFundamentalUnits[Sqrt[2] + Sqrt[3]]
Wolfram Language code: AlgebraicNumberNorm /@ NumberFieldFundamentalUnits[Root[16 - 20 #1^2 + #1^4&, 4]]

Compute the smallest field that includes , i.e. :

Wolfram Language code: e = ToNumberField[{Sqrt[3], Sqrt[-5]}, All][[1, 1]]

Compute the norm in that field:

Wolfram Language code: AlgebraicNumberNorm[Sqrt[3] + Sqrt[-5], Extension -> e]

Neat Examples  (1)

Plot of norms of elements in :

Wolfram Language code: ListPlot[AlgebraicNumberNorm @ Table[AlgebraicNumber[Sqrt[-2] , p], {p, Tuples[Range[-10, 10], 2]}]]
Wolfram Research (2007), AlgebraicNumberNorm, Wolfram Language function, https://reference.wolfram.com/language/ref/AlgebraicNumberNorm.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_algebraicnumbernorm, author="Wolfram Research", title="{AlgebraicNumberNorm}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/AlgebraicNumberNorm.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_algebraicnumbernorm, organization={Wolfram Research}, title={AlgebraicNumberNorm}, year={2007}, url={https://reference.wolfram.com/language/ref/AlgebraicNumberNorm.html}, note=[Accessed: 13-June-2026]}