Cosh[(2 Pi I)/7] == Root[-1 - 4 #1 + 4 #1^2 + 8 #1^3 & , 3]

Cell[BoxData[RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]], "7"], "]"]], "\[Equal]", RowBox[List["Root", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["4", " ", "#1"]], "+", RowBox[List["4", " ", SuperscriptBox["#1", "2"]]], "+", RowBox[List["8", " ", SuperscriptBox["#1", "3"]]]]], "&"]], ",", "3"]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 7 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <msubsup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ; </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> CompoundExpression </ci> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "7"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["Root", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["4", " ", "#1"]], "+", RowBox[List["4", " ", SuperscriptBox["#1", "2"]]], "+", RowBox[List["8", " ", SuperscriptBox["#1", "3"]]]]], "&"]], ",", "3"]], "]"]]]]]]

2001-10-29