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AlgebraicNumberDenominator[a]

gives the smallest positive integer n such that n a is an algebraic integer.

Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
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AlgebraicNumberDenominator

AlgebraicNumberDenominator[a]

gives the smallest positive integer n such that n a is an algebraic integer.

Examples

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Basic Examples  (1)

Compute denominators of simple algebraic numbers:

Wolfram Language code: AlgebraicNumberDenominator[1 / Sqrt[3]]
Wolfram Language code: AlgebraicNumberDenominator[(1 + Sqrt[5]) / 2]

Scope  (3)

Radical expressions:

Wolfram Language code: AlgebraicNumberDenominator[1 / Sqrt[Sqrt[2] + 3]]
Wolfram Language code: AlgebraicNumberDenominator[(1 + 3I) ^ (-1 / 3)]

Root and AlgebraicNumber objects:

Wolfram Language code: AlgebraicNumberDenominator[Root[5 - 6#1 + 3#1 ^ 3 &, 1]]
Wolfram Language code: AlgebraicNumberDenominator[AlgebraicNumber[Sqrt[2], {1 / 5, 1}]]

AlgebraicNumberDenominator automatically threads over lists:

Wolfram Language code: AlgebraicNumberDenominator[{Sqrt[2], 1 / Sqrt[2], 1 / 3}]

Applications  (1)

Representation of 1/(1+) as a quotient α/n of an algebraic integer α and an integer n:

Wolfram Language code: n = AlgebraicNumberDenominator[1 / (1 + Sqrt[3])]
Wolfram Language code: α = n ToNumberField[1 / (1 + Sqrt[3]), Sqrt[3]]
Wolfram Language code: AlgebraicIntegerQ[α]

Properties & Relations  (2)

For an algebraic integer n, the denominator is 1:

Wolfram Language code: AlgebraicNumberDenominator[Sqrt[2]]

Multiplying an algebraic number by its denominator gives an algebraic integer:

Wolfram Language code: α = 1 / Sqrt[1 + I];
Wolfram Language code: AlgebraicIntegerQ[α]
Wolfram Language code: AlgebraicIntegerQ[AlgebraicNumberDenominator[α]α]

Possible Issues  (1)

The argument must be an algebraic number:

Wolfram Language code: AlgebraicNumberDenominator[Pi]
Wolfram Language code: Element[Pi, Algebraics]
Wolfram Research (2007), AlgebraicNumberDenominator, Wolfram Language function, https://reference.wolfram.com/language/ref/AlgebraicNumberDenominator.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_algebraicnumberdenominator, author="Wolfram Research", title="{AlgebraicNumberDenominator}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/AlgebraicNumberDenominator.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_algebraicnumberdenominator, organization={Wolfram Research}, title={AlgebraicNumberDenominator}, year={2007}, url={https://reference.wolfram.com/language/ref/AlgebraicNumberDenominator.html}, note=[Accessed: 12-June-2026]}