因数分解解説

\begin{align} x^{28}+x^{21}+x^{14}+x^7+1 &=\frac{x^{35}-1}{x^7-1}\\ &=\frac{\displaystyle \prod_{k=0}^{34} \left( x-e^{\frac{2k \pi}{35}} \right)}{\displaystyle \prod_{k=0}^{6} \left(x-e^{\frac{2k \pi}{7}}\right)}\\ &=\left(x-e^{i \frac{2 \pi }{35}} \right)\left(x-e^{i \frac{4 \pi }{35}} \right)\left(x-e^{i \frac{6 \pi }{35}} \right)\left(x-e^{i \frac{8 \pi }{35}} \right)\left(x-e^{i \frac{12 \pi }{35}} \right)\\&\ \ \ \ \ \left(x-e^{i \frac{14 \pi }{35}} \right)\left(x-e^{i \frac{16 \pi }{35}} \right)\left(x-e^{i \frac{18 \pi }{35}} \right)\left(x-e^{i \frac{22 \pi }{35}} \right)\left(x-e^{i \frac{24 \pi }{35}} \right)\\&\ \ \ \ \ \left(x-e^{i \frac{26 \pi }{35}} \right)\left(x-e^{i \frac{28 \pi }{35}} \right)\left(x-e^{i \frac{32 \pi }{35}} \right)\left(x-e^{i \frac{34 \pi }{35}} \right)\left(x-e^{i \frac{36 \pi }{35}} \right)\\&\ \ \ \ \ \left(x-e^{i \frac{38 \pi }{35}} \right)\left(x-e^{i \frac{42 \pi }{35}} \right)\left(x-e^{i \frac{44 \pi }{35}} \right)\left(x-e^{i \frac{46 \pi }{35}} \right)\left(x-e^{i \frac{48 \pi }{35}} \right)\\&\ \ \ \ \ \left(x-e^{i \frac{52 \pi }{35}} \right)\left(x-e^{i \frac{54 \pi }{35}} \right)\left(x-e^{i \frac{56 \pi }{35}} \right)\left(x-e^{i \frac{58 \pi }{35}} \right)\left(x-e^{i \frac{62 \pi }{35}} \right)\\&\ \ \ \ \ \left(x-e^{i \frac{64 \pi }{35}} \right)\left(x-e^{i \frac{66 \pi }{35}} \right)\left(x-e^{i \frac{68 \pi }{35}} \right)\\ &=\left(x^2-2\left(\cos\frac{2\pi}{35}\right)x+1\right) \left(x^2-2\left(\cos\frac{4\pi}{35}\right)x+1\right)\left(x^2-2\left(\cos\frac{6\pi}{35}\right)x+1\right)\\&\ \ \ \ \ \left(x^2-2\left(\cos\frac{8\pi}{35}\right)x+1\right) \left(x^2-2\left(\cos\frac{12\pi}{35}\right)x+1\right)\left(x^2-2\left(\cos\frac{14\pi}{35}\right)x+1\right)\\&\ \ \ \ \ \left(x^2-2\left(\cos\frac{16\pi}{35}\right)x+1\right) \left(x^2-2\left(\cos\frac{18\pi}{35}\right)x+1\right)\left(x^2-2\left(\cos\frac{22\pi}{35}\right)x+1\right)\\&\ \ \ \ \ \left(x^2-2\left(\cos\frac{24\pi}{35}\right)x+1\right) \left(x^2-2\left(\cos\frac{26\pi}{35}\right)x+1\right)\left(x^2-2\left(\cos\frac{28\pi}{35}\right)x+1\right)\\&\ \ \ \ \ \left(x^2-2\left(\cos\frac{32\pi}{35}\right)x+1\right) \left(x^2-2\left(\cos\frac{34\pi}{35}\right)x+1\right) \end{align}

$(x^{28}+x^{21}+x^{14}+x^7+1)(x^7-1)=x^{35}-1$に気づくことと,1のn乗根を複素平面でとらえたときに共役な複素数に気づくことができるかどうかが鍵でした。
みゆさん の一般解とフェルマーの最終定理の因数分解のご紹介↓↓↓
みゆさんによる一般解とフェルマーの最終定理の因数分解
ぜひこちらもご覧下さい！