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代數方程<math>x^{7}=1</math>存在李煌根式解

<math>x_7=1</math>

<math>x_1=\frac{ 3(-2548+588\sqrt{3}i)^{\frac{1}{3}}+ \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} -2184+540\sqrt{3}i +168(-2548+588\sqrt{3}i)^{\frac{1}{3}}  }             }{-3(-2548+588\sqrt{3}i)^{\frac{1}{3}}+ \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} -2184+540\sqrt{3}i +168(-2548+588\sqrt{3}i)^{\frac{1}{3}}  }             }</math>

<math>x_2=\frac{ 3(-2548+588\sqrt{3}i)^{\frac{1}{3}}- \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} -2184+540\sqrt{3}i +168(-2548+588\sqrt{3}i)^{\frac{1}{3}}  }             }{-3(-2548+588\sqrt{3}i)^{\frac{1}{3}}- \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} -2184+540\sqrt{3}i +168(-2548+588\sqrt{3}i)^{\frac{1}{3}}  }             }</math>

<math>x_3=\frac{ 3(-2548+588\sqrt{3}i)^{\frac{1}{3}}+ \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} +336-1344\sqrt{3}i -84(-2548+588\sqrt{3}i)^{\frac{1}{3}} -84\sqrt{3}(-2548+588\sqrt{3}i)^{\frac{1}{3}}i  }             }{ -3(-2548+588\sqrt{3}i)^{\frac{1}{3}}+ \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} +336-1344\sqrt{3}i -84(-2548+588\sqrt{3}i)^{\frac{1}{3}} -84\sqrt{3}(-2548+588\sqrt{3}i)^{\frac{1}{3}}i  } }</math>


<math>x_4=\frac{ 3(-2548+588\sqrt{3}i)^{\frac{1}{3}}- \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} +336-1344\sqrt{3}i -84(-2548+588\sqrt{3}i)^{\frac{1}{3}} -84\sqrt{3}(-2548+588\sqrt{3}i)^{\frac{1}{3}}i  }             }{ -3(-2548+588\sqrt{3}i)^{\frac{1}{3}}- \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} +336-1344\sqrt{3}i -84(-2548+588\sqrt{3}i)^{\frac{1}{3}} -84\sqrt{3}(-2548+588\sqrt{3}i)^{\frac{1}{3}}i  } }</math>

<math>x_5=\frac{ 3(-2548+588\sqrt{3}i)^{\frac{1}{3}}+ \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} +1848+840\sqrt{3}i -84(-2548+588\sqrt{3}i)^{\frac{1}{3}} +84\sqrt{3}(-2548+588\sqrt{3}i)^{\frac{1}{3}}i  }             }{ -3(-2548+588\sqrt{3}i)^{\frac{1}{3}}+ \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} +1848+840\sqrt{3}i -84(-2548+588\sqrt{3}i)^{\frac{1}{3}} +84\sqrt{3}(-2548+588\sqrt{3}i)^{\frac{1}{3}}i  } }</math>

<math>x_6=\frac{ 3(-2548+588\sqrt{3}i)^{\frac{1}{3}}- \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} +1848+840\sqrt{3}i -84(-2548+588\sqrt{3}i)^{\frac{1}{3}} +84\sqrt{3}(-2548+588\sqrt{3}i)^{\frac{1}{3}}i  }             }{ -3(-2548+588\sqrt{3}i)^{\frac{1}{3}}- \sqrt{-15(-2548+588\sqrt{3}i)^{\frac{2}{3}} +1848+840\sqrt{3}i -84(-2548+588\sqrt{3}i)^{\frac{1}{3}} +84\sqrt{3}(-2548+588\sqrt{3}i)^{\frac{1}{3}}i  } }</math>

== 研究难度对比 ==
[http://zh.wikipedia.org/wiki/%E4%B8%83%E9%82%8A%E5%BD%A2]

<<[[School:李煌數學研究院]]

== 來源 ==
* 《南昌理工學院學報》.李煌


[[Category:李煌数学研究院]]