试证: $dps{cosfrac{2pi}{5}}$ 为无理数.
证明: 设 $$ex z=e^{ifrac{2pi}{5}}, eex$$ 则 $$eex ea z^5&=e^{i2pi}=1,\ (z-1)(z^4+z^3+z^2+z+1)&=0,\ z^4+z^3+z^2+z+1&=0,\ z^2+z+1+z^{-1}+z^{-2}&=0. eea eeex$$ 比较两端实部, 我们发现 $$eex ea 2cosfrac{4pi}{5}+2cosfrac{2pi}{5}+1&=0,\ 2sex{2cos^2frac{2pi}{5}-1}+2cosfrac{2pi}{5}+1&=0,\ 4cos^2frac{2pi}{5}+2cosfrac{2pi}{5}-1&=0,\ cosfrac{2pi}{5}&=frac{sqrt{5}-1}{4} otinbQ. eea eeex$$