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\({{x}^{101}}=1\)的101個根為\(1,\omega ,{{\omega }^{2}},{{\omega }^{3}},\cdots ,{{\omega }^{100}}\)
\(\begin{align}
  & {{x}^{101}}-1=(x-1)(x-\omega )(x-{{\omega }^{2}})\cdots (x-{{\omega }^{100}}) \\
& {{x}^{100}}+{{x}^{99}}+\cdots +x+1=(x-\omega )(x-{{\omega }^{2}})\cdots (x-{{\omega }^{100}}) \\
& 101=(1-\omega )(1-{{\omega }^{2}})\cdots (1-{{\omega }^{100}})=(\omega -1)({{\omega }^{2}}-1)\cdots ({{\omega }^{100}}-1) \\
\end{align}\)

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