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AlgebraicUnitQ[a]

yields True if a is an algebraic unit, and yields False otherwise.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Basic Applications  
Number Theory  
Properties & Relations  
Possible Issues  
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AlgebraicUnitQ

AlgebraicUnitQ[a]

yields True if a is an algebraic unit, and yields False otherwise.

Details

  • AlgebraicUnitQ is typically used to test whether a number is an algebraic unity or not.
  • An algebraic unit a is a number for which both a and 1/a are algebraic integers.
  • AlgebraicUnitQ[a] returns False unless a is manifestly an algebraic unit.

Examples

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Basic Examples  (2)

Test whether a number is an algebraic unit:

Wolfram Language code: AlgebraicUnitQ[I]

The number is not an algebraic unit:

Wolfram Language code: AlgebraicUnitQ[Sqrt[2]]

Scope  (4)

AlgebraicUnitQ works over integers:

Wolfram Language code: AlgebraicUnitQ[1]

Real numbers:

Wolfram Language code: AlgebraicUnitQ[2.4]

Complex numbers:

Wolfram Language code: AlgebraicUnitQ[Exp[I 2Pi / 5]]

Transcendental numbers:

Wolfram Language code: AlgebraicUnitQ[Pi]

Radical expressions:

Wolfram Language code: AlgebraicUnitQ[5 ^ (2 / 3) + 7 ^ (1 / 5) + 10]

Root objects:

Wolfram Language code: AlgebraicUnitQ[Root[1 + #1 + #1^2 + #1^3 + #1^4 + #1^5 + #1^6&, 1]]

AlgebraicNumber objects:

Wolfram Language code: AlgebraicUnitQ[AlgebraicNumber[Root[#1 ^ 3 - 4 #1 + 17&, 1], {1, 2, 3}]]

AlgebraicUnitQ threads over lists:

Wolfram Language code: AlgebraicUnitQ[{I, Sqrt[2], 1 / Sqrt[5]}]

Applications  (6)

Basic Applications  (1)

Generate random algebraic units:

Wolfram Language code: randomAlgebraicUnit[n_, m_ : 1] := Table[Root[Dot[Table[1, n + 1], # ^ Range[0, n]], RandomInteger[{1, n}]], {m}];
Wolfram Language code: randomAlgebraicUnit[8, 10]
Wolfram Language code: AlgebraicUnitQ[%]

Plot algebraic units:

Wolfram Language code: ComplexListPlot[randomAlgebraicUnit[100, 50]]

Number Theory  (5)

Find integers that are algebraic units:

Wolfram Language code: x /. Solve[x * y == 1, {x, y}, Integers]
Wolfram Language code: {AlgebraicUnitQ[1], AlgebraicUnitQ[-1]}

All roots of unity are algebraic units:

Wolfram Language code: Exp[(2Pi Range[0, 6]I) / 6]
Wolfram Language code: AlgebraicUnitQ[%]

An algebraic unit has norm or :

Wolfram Language code: AlgebraicUnitQ[GoldenRatio]
Wolfram Language code: AlgebraicNumberNorm[GoldenRatio]

Representatives of norm in :

Wolfram Language code: {a} = NumberFieldNormRepresentatives[Sqrt[5], 5]
Wolfram Language code: AlgebraicNumberNorm[a]

It can be represented in terms of the representative a by multiplying by a unit:

Wolfram Language code: AlgebraicUnitQ[-2 - Sqrt[5]]
Wolfram Language code: RootReduce[Sqrt[5](-2 - Sqrt[5]) == a]

Use the roots of unity to find Cyclotomic polynomials:

Wolfram Language code: roots = Exp[2 Pi I / Range[3]]
Wolfram Language code: MinimalPolynomial[roots, x]
Wolfram Language code: Cyclotomic[Range[3], x]

Properties & Relations  (7)

An algebraic unit and its reciprocal are algebraic integers:

Wolfram Language code: a = 1 - 6 6 ^ (1 / 3) + 3 36 ^ (1 / 3);
Wolfram Language code: AlgebraicIntegerQ [{a, 1 / a}]

The reciprocal is an algebraic unit:

Wolfram Language code: AlgebraicUnitQ [{a, 1 / a}]

The only integers that are both algebraic integers and algebraic units are and :

Wolfram Language code: {AlgebraicIntegerQ[1], AlgebraicUnitQ[1]}
Wolfram Language code: {AlgebraicIntegerQ[-1], AlgebraicUnitQ[-1]}

An algebraic unit raised to a power is an algebraic unit:

Wolfram Language code: Table[AlgebraicUnitQ[(1 + Sqrt[2]) ^ n], {n, 5}]

Algebraics represents the domain of all algebraic numbers, including algebraic units:

Wolfram Language code: Element[Exp[Pi I / 2], Algebraics]

An algebraic unit has norm or :

Wolfram Language code: AlgebraicUnitQ[{GoldenRatio, Exp[Pi I / 2]}]
Wolfram Language code: AlgebraicNumberNorm[{GoldenRatio, Exp[Pi I / 2]}]

Use MinimalPolynomial to find the minimal polynomial of an algebraic unit:

Wolfram Language code: MinimalPolynomial[4 / 5, x]

Now find the minimal polynomial of its reciprocal:

Wolfram Language code: MinimalPolynomial[5 / 4, x]

Fundamental units of a number field are algebraic units:

Wolfram Language code: NumberFieldFundamentalUnits[Root[-1 + 2#1 ^ 4&, 1]]
Wolfram Language code: AlgebraicUnitQ[%]

Possible Issues  (1)

In some cases it is not known whether the number is an algebraic unit:

Wolfram Language code: AlgebraicUnitQ[E + Pi]
Wolfram Research (2007), AlgebraicUnitQ, Wolfram Language function, https://reference.wolfram.com/language/ref/AlgebraicUnitQ.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_algebraicunitq, author="Wolfram Research", title="{AlgebraicUnitQ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/AlgebraicUnitQ.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_algebraicunitq, organization={Wolfram Research}, title={AlgebraicUnitQ}, year={2007}, url={https://reference.wolfram.com/language/ref/AlgebraicUnitQ.html}, note=[Accessed: 13-June-2026]}