\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{53} & h_0=\prod_{\sigma_i \in \ V_4}\sigma_i(x-v)=(x-v_1)(x-v_8)(x-v_{17})(x-v_{24}) \\ &h_1=(x-v_4)(x-v_{12})(x-v_{13})(x-v_{21}) \notag \\ &h_2=(x-v_5)(x-v_9)(x-v_{16})(x-v_{20}) \notag \\ \end{align}
\begin{align} \notag \\ &\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} \qquad \begin{array}{l} ( \ Lagrange \ resolvent \ )\\ \\ \quad \bbox[#00FFFF]{ \Omega=\omega^2+\omega+1=0 } \end{array} \\ \notag \\ &\left\{ \begin{array}{l} t_0 \ \in \ F_1[x] \\ \{t_1,t_2\} \ \in \ F_1(v)[x] \end{array} \right. \quad \Longrightarrow \quad \left\{ \begin{array}{l} B_2=a_2^3-A_2=0 \quad A_2 \in F_1 \\ \{\tilde{t_1},\tilde{t_2} \} \ \in \ F_2[x]=F_0(a_1,a_2)[x] \end{array} \right. \\ \notag \\ &\begin{bmatrix} \tilde{h_0} \\ \tilde{h_1} \\ \tilde{h_2} \end{bmatrix} = \begin{bmatrix} 1&1&1 \\ 1&\omega^2&(\omega^2)^2\\ 1&\omega&(\omega^2)\\ \end{bmatrix} \cdot \begin{bmatrix} t_0 \\ \tilde{t_1} \\ \tilde{t_2} \end{bmatrix} \quad \Longrightarrow \quad \left\{ \begin{array}{l} g_1(x)=\tilde{h_0}\cdot \tilde{h_1} \cdot \tilde{h_2} \\ g_2(x) \equiv \tilde{h_0} \ \in \ F_2[x] \end{array} \right. \\ \notag \\ & g_1(x)=0 \quad \Rightarrow \quad \left\{ \begin{array}{l} g_2(x)=0\\ B_2=0 \end{array} \right. \\ \end{align}
\begin{align} & t_0=x^4+112 \\ \end{align}
\begin{align} & t_1=\Biggl[\frac{176356098785 {a_1} {{v}^{10}} \omega }{26343508540135700992}+...+\frac{27924690836995920}{2708008690392239} \Biggr]x^2 +.... \\ & t_2=\Biggl[ -\frac{176356098785 {a_1} {{v}^{10}} \omega }{26343508540135700992}+...-\frac{193463308578480}{2708008690392239} \Biggr]x^2 +.... \\ \end{align}
\begin{align} a_2=&\Biggl[\frac{176356098785 {a_1} {{v}^{10}} \omega }{26343508540135700992} +\frac{103314391395 {{v}^{10}} \omega }{5416017380784478}+.....+\frac{27924690836995920}{2708008690392239} \Biggr] \notag \\ b_2=&\Biggl[ -\frac{176356098785 {a_1} {{v}^{10}} \omega }{26343508540135700992} -\frac{103314391395 {{v}^{10}} \omega }{5416017380784478}+.....-\frac{193463308578480}{2708008690392239} \Biggr] \notag \\ a_2^{-1}=&\Biggl[-\frac{529068296355 {a_1} {{v}^{10}} \omega }{10958899552696451612672} -\frac{309943174185 {{v}^{10}} \omega }{2253063230406342848}+.....-\frac{36274370358465}{70408225950198214} \Biggr] \notag \\ b_2^{-1}=&\Biggl[\frac{529068296355 {a_1} {{v}^{10}} \omega }{10958899552696451612672} +\frac{309943174185 {{v}^{10}} \omega }{2253063230406342848}+.....+\frac{5235879531936735}{70408225950198214} \Biggr] \notag \\ \notag \\ \end{align}
\begin{align} q_1= \ a_2^{-1} \cdot t_1= x^2-\frac{5 {a_1} \omega }{2496}+\frac{18 \omega }{13}-\frac{31 {a_1}}{24960}-\frac{207}{26} \ \in \ F_1[x] \\ q_2= \ b_2^{-1} \cdot t_2= x^2+\frac{5 {a_1} \omega }{2496}-\frac{18 \omega }{13}+\frac{19 {a_1}}{24960}-\frac{243}{26} \ \in \ F_1[x] \notag \\ \end{align}
\begin{align} &a_2^3= \frac{14 {a_1} \omega }{27}+2304 \omega +\frac{89 {a_1}}{135}-1088 \equiv A_2 \ \in \ F_1\\ \notag \\ &B_2 \equiv a_2^3-A_2=0 \quad a_2 \equiv \sqrt[3]{A_2} \quad \Rightarrow \quad \bbox[#FFFF00]{ F_2 \equiv F_1(a_2) } \\ \end{align}
\begin{align} & A_2^{-1}=-\frac{7 {a_1} \omega }{35995648}-\frac{243 \omega }{281216}+\frac{19 {a_1}}{359956480}-\frac{1431}{1124864} \in F_1 \\ &a_2 \cdot b_2=\frac{416}{3} \ \in \ F_0\\ \notag \\ &b_2=\frac{a_2^2 \cdot (a_2 \cdot b_2)}{a_2^3}=A_2^{-1} \cdot a_2^2 \cdot (a_2 \cdot b_2) \notag \\ &\quad \therefore \ b_2=a_2^2\biggl(-\frac{7 {a_1} \omega }{259584}-\frac{81 \omega }{676}+\frac{19 {a_1}}{2595840}-\frac{477}{2704}\biggr) \\ \notag \\ &\tilde{t_1}=a_2 \cdot q_1=a_2\biggl( x^2-\frac{5 {a_1} \omega }{2496}+\frac{18 \omega }{13}-\frac{31 {a_1}}{24960}-\frac{207}{26} \biggr) \ \in \ F_2[x]\\ &\tilde{t_2}=b_2 \cdot q_2=b_2\biggl( x^2+\frac{5 {a_1} \omega }{2496}-\frac{18 \omega }{13}+\frac{19 {a_1}}{24960}-\frac{243}{26} \biggr) \ \in \ F_2[x]\\ \end{align}
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