\begin{align*} &f(x)=x^4+4x+2 \\ &\qquad \{\alpha,\beta,\gamma,\delta\}: \ roots \ of \ f(x)\\ \\ & v: \ Primitive \ element \\ & \qquad v=1\cdot\alpha+2\cdot\beta+3\cdot\gamma+4 \cdot \delta \end{align*}
\[ \qquad The \ system \ of \ equations \]
\[ \left\{ \begin{array}{l} r_1={{\alpha }^{4}}+4\alpha +2=0\\ r_2={{\beta }^{3}}+\alpha {{\beta }^{2}}+{{\alpha }^{2}} \beta +{{\alpha }^{3}}+4=0 \\ r_3={{\gamma }^{2}}+\left( \beta +\alpha \right) \gamma +{{\beta }^{2}}+\alpha \beta +{{\alpha }^{2}}=0\\ r_4= \alpha+\beta+\gamma+\delta =0\\ r_5=v-(\alpha+2\beta+3\gamma+4\delta )=0 \\ \end{array} \right.\\ \quad \\ \qquad \qquad \qquad \Downarrow \]
\[ \qquad Elimination \ Theory \]
\[ V(v)=v^{24}-160v^{20}+5440v^18+30080v^{16}+...\\ \quad...+700091596800v^2+4691625312256\\ \]
\begin{align*} &V(x): \ irreducible \ polynomial \\ \\ &\therefore \ g_0(x) \equiv V(x) \qquad deg(g_0(x))=24\\ \\ &g_0(x):minimal \ polynomial \ of \ v \ on \ F_0=Q(\omega) \\ \end{align*}
\[Factorization \ of \ f(x) \ on \ F_0(v)\] \[\quad "maxima's \ function \ "\] \[\qquad factor(f(x),g_0(v))\]
\begin{align*} &\alpha=\alpha(v), \ \beta=\beta(v), \ \gamma=\gamma(v), \ \delta=\delta(v)\\ \\ &roots \ of \ g_0(x) \ ( \ =V(x) \ )\\ &\quad [ \ v_1=v_1(v), \ ....\ , \ v_{24}=v_{24}(v) \ ] \\ \end{align*}
\begin{align*} &S_4: Galois \ group \ of \ f(x) \\ &composition \ series \quad S_4 \rhd \ A_4 \rhd \ V_4 \rhd \{e\} \end{align*}
\[g_1(x)\ : \ minimal \ polynomial \ of \ v\\ \qquad g_1(x) \ \in \ F_0(a_1)[x]\qquad deg(g_1(x))=12 \\ \quad \\ B_1=a_1^2+17510400\]
\[g_2(x)\ : \ minimal \ polynomial \ of \ v\\ \qquad g_2(x) \ \in \ F_1(a_2)[x] \qquad deg(g_2(x))=4\\ \quad \\ B_2=a_2^3-\frac{14 {a_1} \omega }{27}-2304 \omega -\frac{89 {a_1}}{135}+1088 \]
\[ g_3(x)\ : \ minimal \ polynomial \ of \ v\\ \qquad g_3(x) \ \in \ F_2(a_3)[x] \qquad deg(g_3(x))=2\\ \]
\[ B_3=a_3^2-\biggl( \frac{23 {a_1} {{a}_{2}^{2}} \omega }{324480}+\frac{63 {{a}_{2}^{2}} \omega }{338}-\frac{8 {a_2} \omega }{13}\\ \qquad \qquad +\frac{{a_1} {{a}_{2}^{2}}}{324480}+\frac{135 {{a}_{2}^{2}}}{338}-\frac{32 {a_2}}{13} \biggr)\\ \]
\[ g_4(x)\ : \ minimal \ polynomial \ of \ v\\ \qquad g_4(x) \ \in \ F_3(a_4)[x] \qquad deg(g_4(x))=1\\ \]
\[ B_4=a_4^2-\biggl(-\frac{11 {a_1} {{a}_{2}^{2}} \omega }{2595840}+\frac{9 {{a}_{2}^{2}} \omega }{676}+\frac{2 {a_2} \omega }{13} \\ \qquad \qquad -\frac{23 {a_1} {{a}_{2}^{2}}}{5191680}-\frac{63 {{a}_{2}^{2}}}{5408}+\frac{3 {a_2}}{26}\biggr)\\ \]
\begin{align*} &v=v(a_1,a_2,a_3,a_4,\omega) \ \in \ F_4=F_0(a_1,a_2,a_3,a_4,\omega) \\ \\ &\left\{ \begin{array}{l} \alpha=\alpha(a_1,a_2,a_3,a_4,\omega), \ \ \beta=\beta(a_1,a_2,a_3,a_4,\omega) \\ \gamma=\gamma(a_1,a_2,a_3,a_4,\omega), \ \ \delta=\delta(a_1,a_2,a_3,a_4,\omega) \\ \end{array} \right.\\ \end{align*}
\begin{align} \setCounter{72} & h_0=\prod_{\sigma_i \in \ N}\sigma_i(x-v)=(x-v_1)(x-v_{8}) \\ &h_1=\prod_{\sigma_i \in \ (V_4-N)}\sigma_i(x-v)=(x-v_{17})(x-v_{24}) \notag \\ \end{align}
\begin{align} \notag \\ & \begin{bmatrix} t_0 \\ t_1 \end{bmatrix} =\frac{1}{2} \begin{bmatrix} 1&1 \\ 1&-1 \end{bmatrix} \cdot \begin{bmatrix} h_0 \\ h_1 \end{bmatrix} \qquad ( \ Lagrange \ resolvent \ )\\ \notag \\ &\left\{ \begin{array}{l} t_0 \ \in \ F_2[x] \\ t_1 \ \in \ F_2(v)[x] \end{array} \right. \quad \Longrightarrow \quad \left\{ \begin{array}{l} B_3=a_3^2-A_3=0 \quad A_3 \in F_2 \\ \tilde{t_1} \ \in \ F_3[x]=F_0(a_1,a_2,a_3)[x] \end{array} \right. \\ \notag \\ &\begin{bmatrix} \tilde{h_0} \\ \tilde{h_1 } \end{bmatrix} = \begin{bmatrix} 1&1 \\ 1&-1 \end{bmatrix} \cdot \begin{bmatrix} t_0 \\ \tilde{t_1} \end{bmatrix} \quad \Longrightarrow \quad \left\{ \begin{array}{l} g_2(x)=\tilde{h_0} \cdot \tilde{h_1} \\ g_3(x) \equiv \tilde{h_0} \ \in \ F_3[x] \end{array} \right. \\ \notag \\ & g_2(x)=0 \quad \Rightarrow \quad \left\{ \begin{array}{l} g_3(x)=0\\ B_3=0 \end{array} \right. \\ \end{align}
\begin{align} & t_0= x^2+\frac{19 {a_1} {{a}_{2}^{2}} \omega }{865280}+\frac{45 {{a}_{2}^{2}} \omega }{1352}-\frac{4 {a_2} \omega }{13}+\frac{9 {a_1} {{a}_{2}^{2}}}{1730560}+\frac{603 {{a}_{2}^{2}}}{5408}-\frac{19 {a_2}}{26} \\ & t_1=\Biggl[-\frac{5 {a_1} {{a}_{2}^{2}} {{v}^{3}} \omega }{183028352}+\frac{1215 {{a}_{2}^{2}} {{v}^{3}} \omega }{5719636} +.....+\frac{49 {a_1} v}{338440}-\frac{21371 v}{8461}\Biggr]x \\ \end{align}
\begin{align} a_3=&\Biggl[ -\frac{5 {a_1} {{a}_{2}^{2}} {{v}^{3}} \omega }{183028352}+\frac{1215 {{a}_{2}^{2}} {{v}^{3}} \omega }{5719636} +.....+\frac{49 {a_1} v}{338440}-\frac{21371 v}{8461} \Biggr] \ \in \ F_2(v) \\ a_3^{-1}=&\Biggl[\frac{46613 {a_1} {{a}_{2}^{2}} {{v}^{3}} \omega }{11245261946880}+\frac{81867 {{a}_{2}^{2}} {{v}^{3}} \omega }{11713814528} +....-\frac{7 {a_1} v}{1624512}+\frac{18755 v}{812256} \Biggr] \\ \end{align}
\begin{align} q_1= \ a_3^{-1} \cdot t_1= x \quad \in \ F_2[x] \\ \end{align}
\begin{align} &a_3^2=\frac{23 {a_1} {{a}_{2}^{2}} \omega }{324480}+\frac{63 {{a}_{2}^{2}} \omega }{338}-\frac{8 {a_2} \omega }{13}+\frac{{a_1} {{a}_{2}^{2}}}{324480}+\frac{135 {{a}_{2}^{2}}}{338}-\frac{32 {a_2}}{13} \equiv A_3 \ \in \ F_2\\ \notag \\ \end{align}
\begin{align} &B_3 \equiv a_3^2-A_3=0 \quad a_3 \equiv \sqrt{A_3} \quad \Rightarrow \quad \bbox[#FFFF00]{ F_3 \equiv F_2(a_3) } \\ \notag \\ &\tilde{t_1}=a_3 \cdot q_1=a_3x \ \in \ F_3[x]\\ \end{align}
最後に、式(78)の \(t_0\) と式(85)の \(\{ \ \tilde{t_1} \ \}\) を、式(76)に代入すると、 \(F_3\) 上の \(v\) の最小多項式 \(\tilde{h_0} \equiv g_3(x)\) を求める事が出来ます。 以上、計算結果をまとめると以下の様になります。\begin{align} &\tilde{h_0}=t_0+\tilde{t_1} \equiv g_3(x) \\ &\qquad \Downarrow \notag \\ &\ g_3(x)=x^2+ \frac{19 {a_1} {{a}_{2}^{2}} \omega }{865280}+\frac{45 {{a}_{2}^{2}} \omega }{1352}-\frac{4 {a_2} \omega }{13}\notag \\ &\qquad \qquad +\frac{9 {a_1} {{a}_{2}^{2}}}{1730560}+\frac{603 {{a}_{2}^{2}}}{5408}-\frac{19 {a_2}}{26}+{a_3} x \ \in \ F_3[x]\\ \notag \\ &\left\{ \begin{array}{l} g_3(v)=0 \\ B_3= \ a_3^2-A_3=0 \qquad \therefore \ a_3=\sqrt{A_3} \qquad F_3 \equiv F_2(a_3)\\ \end{array} \right. \\ \notag \\ \end{align}
\begin{align} &A_3=\frac{23 {a_1} {{a}_{2}^{2}} \omega }{324480}+\frac{63 {{a}_{2}^{2}} \omega }{338}-\frac{8 {a_2} \omega }{13}+\frac{{a_1} {{a}_{2}^{2}}}{324480}+\frac{135 {{a}_{2}^{2}}}{338}-\frac{32 {a_2}}{13} \\ \end{align}
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