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Cyclotomic






Mathematica Notation

Traditional Notation









Polynomials > Cyclotomic[n,z] > Products > Finite products





http://functions.wolfram.com/05.11.24.0002.01









  


  










Input Form





DivisorProduct[Cyclotomic[d, z], {d, 2 n}]/DivisorProduct[Cyclotomic[d, z], {d, n}] == z^n + 1 /; Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[FractionBox[RowBox[List["DivisorProduct", "[", RowBox[List[RowBox[List["Cyclotomic", "[", RowBox[List["d", ",", "z"]], "]"]], ",", RowBox[List["{", RowBox[List["d", ",", RowBox[List["2", "n"]]]], "}"]]]], "]"]], RowBox[List["DivisorProduct", "[", RowBox[List[RowBox[List["Cyclotomic", "[", RowBox[List["d", ",", "z"]], "]"]], ",", RowBox[List["{", RowBox[List["d", ",", "n"]], "}"]]]], "]"]]], "\[Equal]", RowBox[List[SuperscriptBox["z", "n"], "+", "1"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mstyle scriptlevel='0' minsize='2.4'> <munder> <mo> &#8719; </mo> <mrow> <mi> d </mi> <mo> &#8739; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> </munder> </mstyle> <mrow> <mstyle scriptlevel='0'> <msub> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, Cyclotomic] </annotation> </semantics> <mi> d </mi> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> <mrow> <mstyle scriptlevel='0' minsize='2.4'> <munder> <mo> &#8719; </mo> <mrow> <mi> d </mi> <mo> &#8739; </mo> <mi> n </mi> </mrow> </munder> </mstyle> <mrow> <mstyle scriptlevel='0'> <msub> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, Cyclotomic] </annotation> </semantics> <mi> d </mi> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mfrac> <mstyle scriptlevel='0'> <mo> &#10869; </mo> </mstyle> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8719; </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> d </ms> <ms> &#8739; </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> C </ms> <ci> Cyclotomic </ci> </apply> <ms> d </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> d </ms> <ms> &#8739; </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> C </ms> <ci> Cyclotomic </ci> </apply> <ms> d </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> n </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#8712; </ms> <ms> &#8469; </ms> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List["DivisorProduct", "[", RowBox[List[RowBox[List["Cyclotomic", "[", RowBox[List["d_", ",", "z_"]], "]"]], ",", RowBox[List["{", RowBox[List["d_", ",", RowBox[List["2", " ", "n_"]]]], "}"]]]], "]"]], RowBox[List["DivisorProduct", "[", RowBox[List[RowBox[List["Cyclotomic", "[", RowBox[List["d_", ",", "z_"]], "]"]], ",", RowBox[List["{", RowBox[List["d_", ",", "n_"]], "}"]]]], "]"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", "n"], "+", "1"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29