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RootOfUnityQ
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RootOfUnityQ

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yields True if a is a root of unity, and yields False otherwise.

Details

  • An algebraic number a is a root of unity if an=1 for some integer n.

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-zdrcn
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0v5zw1r0qshy7-ezux9n
Out[2]=2

Scope  (5)Survey of the scope of standard use cases

Radical expressions:

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-ftwumt
Out[1]=1

Root objects:

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-uuo64p
Out[1]=1

AlgebraicNumber objects:

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-jz69lg
Out[1]=1

Transcendental objects:

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-mk81wz
Out[1]=1

RootOfUnityQ threads automatically over lists:

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-v8wun2
Out[1]=1

Properties & Relations  (4)Properties of the function, and connections to other functions

Roots of unity are solutions of for some integer n:

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-57v67s
In[2]:=2
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https://wolfram.com/xid/0v5zw1r0qshy7-7vw1cj
Out[2]=2

All roots of unity are algebraic integers that lie on the unit circle:

In[3]:=3
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https://wolfram.com/xid/0v5zw1r0qshy7-zjlipo
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0v5zw1r0qshy7-c36xrn
Out[4]=4

Not all algebraic numbers on the unit circle are roots of unity:

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-ypm
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0v5zw1r0qshy7-m6b
Out[2]=2

The minimal polynomial of a root of unity is a cyclotomic polynomial or one of its factor:

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-0mxkoy
In[2]:=2
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https://wolfram.com/xid/0v5zw1r0qshy7-ixr85k
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0v5zw1r0qshy7-6i073w
Out[3]=3

Roots of cyclotomic polynomials are roots of unity:

In[4]:=4
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https://wolfram.com/xid/0v5zw1r0qshy7-fmhqim
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0v5zw1r0qshy7-fhx8f2
Out[5]=5

Use NumberFieldRootsOfUnity to find all roots of unity in a number field:

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-4a3egw
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0v5zw1r0qshy7-ptssnu
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

Approximate numbers will always return False:

In[1]:=1
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https://wolfram.com/xid/0v5zw1r0qshy7-g46gtl
Out[1]=1

Use RootApproximant to get an exact number:

In[2]:=2
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https://wolfram.com/xid/0v5zw1r0qshy7-b3x3b4
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0v5zw1r0qshy7-iz5src
Out[3]=3
Wolfram Research (2007), RootOfUnityQ, Wolfram Language function, https://reference.wolfram.com/language/ref/RootOfUnityQ.html.
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Wolfram Research (2007), RootOfUnityQ, Wolfram Language function, https://reference.wolfram.com/language/ref/RootOfUnityQ.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_rootofunityq, author="Wolfram Research", title="{RootOfUnityQ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RootOfUnityQ.html}", note=[Accessed: 04-April-2025 ]}

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@misc{reference.wolfram_2025_rootofunityq, author="Wolfram Research", title="{RootOfUnityQ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RootOfUnityQ.html}", note=[Accessed: 04-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_rootofunityq, organization={Wolfram Research}, title={RootOfUnityQ}, year={2007}, url={https://reference.wolfram.com/language/ref/RootOfUnityQ.html}, note=[Accessed: 04-April-2025 ]}

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@online{reference.wolfram_2025_rootofunityq, organization={Wolfram Research}, title={RootOfUnityQ}, year={2007}, url={https://reference.wolfram.com/language/ref/RootOfUnityQ.html}, note=[Accessed: 04-April-2025 ]}