School:李煌數學研究院/紀念偉大的愛情主義悲劇梵高

代數方程$x^7=1$存在李煌根式解

$x_7=1$

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

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

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

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

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

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

研究难度对比

[1]

<<School:李煌數學研究院

來源

• 《南昌理工學院學報》.李煌