({{, (RefLink[NumberFieldRootsOfUnity, paclet:ref/NumberFieldRootsOfUnity])[a] 
gives the roots of unity for the field ℚ[a] generated by the algebraic number a.}})

OpenersNotesSection(Spacer1)Details

(RefLink[NumberFieldRootsOfUnity, paclet:ref/NumberFieldRootsOfUnity]) gives a list of all algebraic numbers (Subscript[u, i]) in the field ℚ[a] for which (Subscript[u, i]^n)=1 for some n.

(RefLink[NumberFieldRootsOfUnity, paclet:ref/NumberFieldRootsOfUnity]) returns any (Subscript[u, i]) other than ±1 as (RefLink[AlgebraicNumber, paclet:ref/AlgebraicNumber]) objects in the field ℚ[a].

OpenersPrimaryExamplesSection(Spacer1)Examples  (8)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

NumberFieldRootsOfUnity[I]

RootReduce[%]

NumberFieldRootsOfUnity[I+2^(1/3)]

RootReduce[%]

OpenersExampleSection1(Spacer1)Scope  (4)

Radical expressions:

NumberFieldRootsOfUnity[Sqrt[-3]]

(RefLink[Root, paclet:ref/Root]) objects:

NumberFieldRootsOfUnity[Root[9-2 #1^2+#1^4&,1]]

(RefLink[AlgebraicNumber, paclet:ref/AlgebraicNumber]) objects:

NumberFieldRootsOfUnity[AlgebraicNumber[Root[#^4-#^2+1&,1],{1,2,3,1}]]

(RefLink[NumberFieldRootsOfUnity, paclet:ref/NumberFieldRootsOfUnity]) automatically threads over lists:

NumberFieldRootsOfUnity[{Sqrt[2], I 2^(1/3)}]

OpenersExampleSection3(Spacer1)Properties & Relations  (2)

A list of roots of unity in the field ℚ[1+i(Sqrt[3])]:

roots=NumberFieldRootsOfUnity[1+ I Sqrt[3]]

Roots of unity are also algebraic integers and units:

AlgebraicIntegerQ[roots]

AlgebraicUnitQ[roots]

And they all have norm ±1:

AlgebraicNumberNorm[roots]

Use (RefLink[RootReduce, paclet:ref/RootReduce]) to get canonical complex expressions:

NumberFieldRootsOfUnity[I+2^(1/3)]

RootReduce[%]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[NumberFieldFundamentalUnits, paclet:ref/NumberFieldFundamentalUnits])  ▪  (RefLink[RootOfUnityQ, paclet:ref/RootOfUnityQ])  ▪  (RefLink[RootReduce, paclet:ref/RootReduce])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Algebraic Number Fields, paclet:tutorial/AlgebraicNumberFields])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Algebraic Number Theory, paclet:guide/AlgebraicNumberTheory])}})}})