APX6　拡大体\(F_i\)での計算の注意点

\begin{align} \setCounter{0} &\begin{bmatrix} t_0 \\ t_1 \\ t_2 \end{bmatrix} =\frac{1}{3} \begin{bmatrix} 1&1&1 \\ 1&\omega&\omega^2\\ 1&(\omega^2)&(\omega^2)^2\\ \end{bmatrix} \cdot \begin{bmatrix} h_0 \\ h_1 \\ h_2 \end{bmatrix} \\ \notag \\ &\therefore \quad \left \{ \begin{array}{l} t_0=x^4+112 \\ t_1=a_2x^2+a_0 \\ t_2=b_2x^2+b_0 \\ \end{array} \right. \ \Rightarrow \ \left \{ \begin{array}{l} t_1=a_2\biggl(x^2+\frac{a_0}{a_2}\biggr) \\ t_2=b_2\biggl(x^2+\frac{b_0}{b_2}\biggr) \\ \end{array} \right.\\ &\qquad \qquad \{ a_2,a_0, b_2,b_0 \} \in \ F_1(v) \notag \\ \end{align}

\begin{align} &\left\{ \begin{array}{l} g_1(v)={{v}^{12}}-80 {{v}^{8}}+{a_1} \left( {{v}^{6}}-\frac{232 {{v}^{2}}}{5}+432\right) +2720 {{v}^{6}} +11840 {{v}^{4}} \\ \qquad+587520 {{v}^{2}}+1193216=0 \\ B_1=a_1^2+17510400=0\\ \Omega=\omega^2+\omega+1=0\\ \end{array} \right.\\ \notag \\ & \{1,v^1,v^2,....,v^{10},v^{11}\}, \quad \{1,a_1\}, \quad \{1,\omega\} \\ &\qquad \qquad \Downarrow \notag \\ & \{1,v^1,v^2,....,v^{10},v^{11}\}\otimes \{1,a_1\}\otimes\{1,\omega\} \notag \\ \notag \\ & \therefore \ Basis \ of \ F_1(v) \quad \bbox[#FFFF00]{ \{v^i\cdot a_1^j\cdot \omega^k\} } \\ & \qquad \qquad \bigl(i=[0,1,...,10,11] \quad j=[0,1] \quad k=[0,1]\bigr) \notag \\ \end{align}

\begin{align} &(a_2)^{-1}=K_1(v,a_1,\omega)= \displaystyle \sum_{i,j,k} c_{ijk} \cdot v^i \cdot a_1^j \cdot \omega^k \\ &(b_2)^{-1}=K_2(v,a_1,\omega)= \displaystyle \sum_{i,j,k} d_{ijk} \cdot v^i \cdot a_1^j \cdot \omega^k \\ \notag \\ &\bbox[#FFFF00]{ a_2 \cdot (a_2)^{-1}=1, \quad b_2 \cdot (b_2)^{-1}=1 } \quad \rightarrow \quad system \ of \ equations \\ \end{align}