节 1.6 复系数、实系数多项式的标准分解式

\begin{equation*} \alpha_k=a\left(\cos\frac{2k\pi}{n}+{\rm i}\sin\frac{2k\pi}{n}\right),k=0,1,\cdots ,n-1, \end{equation*}

\begin{equation*} f(x)=(x- a)(x- \alpha_1)\cdots (x- \alpha_{n-1}). \end{equation*}

\begin{equation*} x^n-1=(x-1)(x- \omega_1)(x- \omega_2)\cdots (x- \omega_{n-1}). \end{equation*}

\begin{equation} x^{n-1}+x^{n-2}+\cdots+1=(x- \omega_1)\cdots (x- \omega_{n-1}).\tag{1.6.1} \end{equation}

\begin{equation*} \omega_k=\cos\frac{k\pi}{3}+{\rm i}\sin\frac{k\pi}{3},k=\pm 1,\pm 2, \end{equation*}

\begin{equation*} x^4+x^2+1|f_1(x^3)+x^4f_2(x^3), \end{equation*}

\begin{equation*} x-\omega_k|f_1(x^3)+x^4f_2(x^3),k=\pm 1,\pm 2, \end{equation*}

\begin{equation*} \left\{ \begin{array}{l} f_1(\omega_{-1}^3)+\omega_{-1}^4f_2(\omega_{-1}^3)=0,\\ f_1(\omega_1^3)+\omega_1^4f_2(\omega_1^3)=0,\\ f_1(\omega_{-2}^3)+\omega_{-2}^4f_2(\omega_{-2}^3)=0,\\ f_1(\omega_2^3)+\omega_2^4f_2(\omega_2^3)=0.\\ \end{array}\right. \end{equation*}

\begin{align} f_1(-1)-\omega_{-1}f_2(-1)=0,&f_1(-1)-\omega_1f_2(-1)=0,\tag{1.6.2}\\ f_1(1)+\omega_{-2}f_2(1)=0,&f_1(1)+\omega_2f_2(1)=0.\tag{1.6.3} \end{align}

\begin{equation*} f_1(-1)=f_2(-1)=0. \end{equation*}

\begin{equation*} f_1(1)=f_2(1)=0. \end{equation*}

\begin{equation*} x+1|\left(f_1(x),f_2(x)\right),x-1|\left(f_1(x),f_2(x)\right). \end{equation*}

\begin{equation*} (x+1)(x-1)|\left(f_1(x),f_2(x)\right). \end{equation*}

\begin{equation*} \deg \left(f_1(x),f_2(x)\right)\leq 2. \end{equation*}

\begin{equation*} \left(f_1(x),f_2(x)\right)=x^2-1. \end{equation*}

\begin{equation*} f(x)=\left(x-(a+d{\rm i})\right)\left(x-(a-d{\rm i})\right)=(x-a)^2+d^2, \end{equation*}

\begin{equation*} g(x)=x-a,h(x)=d, \end{equation*}

\begin{equation*} f(x)=\left(g_1^2(x)+h_1^2(x)\right)\left(g_2^2(x)+h_2^2(x)\right), \end{equation*}

\begin{equation*} \begin{array}{cl} &\left(g_1^2(x)+h_1^2(x)\right)\left(g_2^2(x)+h_2^2(x)\right)\\=&g_1^2(x)g_2^2(x)+g_1^2(x)h_2^2(x)+h_1^2(x)g_2^2(x)+h_1^2(x)h_2^2(x)\\ =&\left(g_1(x)g_2(x)+h_1(x)h_2(x)\right)^2+\left(g_1(x)h_2(x)-h_1(x)g_2(x)\right)^2, \end{array} \end{equation*}

\begin{equation*} g(x)=g_1(x)g_2(x)+h_1(x)h_2(x),h(x)=g_1(x)h_2(x)-h_1(x)g_2(x), \end{equation*}

\begin{equation*} f(x)=g^2(x)+h^2(x) \end{equation*}

\begin{align*} \deg h(x)&=\deg\left(g_1(x)h_2(x)-h_1(x)g_2(x)\right) \\ &\leq\max\{\deg\left(g_1(x)h_2(x)\right),\deg\left(h_1(x)g_2(x)\right)\} \\ &<\deg g_1(x)+\deg g_2(x) \\ &=\deg g(x). \end{align*}

\begin{equation*} \omega_k=\cos\frac{(2k+1)\pi}{n}+{\rm i}\sin\frac{(2k+1)\pi}{n},k=0,1,\cdots ,n-1, \end{equation*}

\begin{equation*} f(x)=(x- \omega_0)(x- \omega_1)\cdots (x- \omega_{n-1}). \end{equation*}

\begin{align*} \overline{\omega_k}&=\cos\frac{(2k+1)\pi}{n}-{\rm i}\sin\frac{(2k+1)\pi}{n} \\ &=\cos\left[2\pi-\frac{(2k+1)\pi}{n}\right]+{\rm i}\sin\left[2\pi-\frac{(2k+1)\pi}{n}\right] \\ &=\cos\frac{\left[2(n-k-1)+1\right]\pi}{n}-{\rm i}\sin\frac{\left[2(n-k-1)+1\right]\pi}{n} \\ &=\omega_{n-k-1}. \end{align*}

\begin{equation*} c_1+c_2+c_3=-p,\ c_1c_2+c_1c_3+c_2c_3=q, \end{equation*}

\begin{equation*} \begin{array}{ccl} p^2-3q&=&(c_1+c_2+c_3)^2-3(c_1c_2+c_1c_3+c_2c_3)\\&=&c_1^2+c_2^2+c_3^2-c_1c_2-c_1c_3-c_2c_3\\&=&\frac{1}{2}\left[(c_1-c_2)^2+(c_1-c_3)^2+(c_2-c_3)^2\right]\geq 0, \end{array} \end{equation*}

\begin{equation*} \left\{\begin{array}{l} \alpha+\beta+\gamma=-a,\\\alpha \beta+\alpha \gamma+\beta \gamma=b,\\\alpha \beta \gamma=-c, \end{array}\right. \end{equation*}

\begin{align*} m&=-\left(\frac{\alpha}{\beta \gamma}+\frac{\beta}{\alpha \gamma}+\frac{\gamma}{\alpha \beta}\right) \\ &=-\frac{\alpha^2+\beta^2+\gamma^2}{\alpha \beta \gamma} \\ &=-\frac{(\alpha+\beta+\gamma)^2-2(\alpha \beta+\alpha \gamma+\beta \gamma)}{\alpha \beta \gamma} \\ &=\frac{a^2-2b}{c}, \end{align*}

\begin{align*} n&=\frac{\alpha}{\beta \gamma}\cdot\frac{\beta}{\alpha \gamma}+\frac{\alpha}{\beta \gamma}\cdot\frac{\gamma}{\alpha \beta}+\frac{\beta}{\alpha \gamma}\cdot\frac{\gamma}{\alpha \beta} \\ &=\frac{\alpha^2 \beta^2+\alpha^2 \gamma^2+\beta^2 \gamma^2}{\alpha^2 \beta^2 \gamma^2} \\ &=\frac{(\alpha \beta+\beta \gamma+\alpha \gamma)^2-2 \alpha \beta \gamma(\alpha+\beta+\gamma)}{(\alpha\beta\gamma)^2} \\ &=\frac{b^2-2ac}{c^2}, \end{align*}

\begin{equation*} p=-\left(\frac{\alpha}{\beta \gamma}\cdot\frac{\beta}{\alpha \gamma}\cdot\frac{\gamma}{\alpha \beta}\right) =-\frac{1}{\alpha \beta \gamma}=\frac{1}{c}, \end{equation*}

\begin{equation*} x^3+\frac{a^2-2b}{c}x^2+\frac{b^2-2ac}{c^2}x+\frac{1}{c}. \end{equation*}

\begin{equation*} f(x)=dp_1^{e_1}(x)\cdots p_m^{e_m}(x), \end{equation*}

\begin{equation*} h(x)=\left[f(x)-g(x)\right]f'(x), \end{equation*}

\begin{equation} \deg h(x)\leq 2n-1.\tag{1.6.5} \end{equation}

\begin{equation*} f(\omega_i)-g(\omega_i)=0-0=0, \end{equation*}

\begin{equation*} \sum\limits_{i=1}^s k_i+\sum\limits_{i=1}^t m_i=2n. \end{equation*}

\begin{equation*} h(x)=0. \end{equation*}

\begin{equation*} \widetilde{f}(x)=a_0x^5+a_3x^2+a_4x+1 \end{equation*}

\begin{equation*} b_1+\cdots+b_5=0,\ \sum\limits_{1\leq i< j\leq 5}b_ib_j=0, \end{equation*}

\begin{equation*} b_1^2+\cdots +b_5^2=(b_1+\cdots+b_5)^2-2\left(\sum\limits_{1\leq i< j\leq 5}b_ib_j\right)=0, \end{equation*}

\begin{equation*} f(x)=a_nx^n+\cdots+a_1x+a_0, \end{equation*}

\begin{equation*} f'(x)=na_nx^{n-1}+\cdots +2a_2x+a_1 \end{equation*}