\begin{align*} &f(x)=3x^3-3x+1 \quad \{\alpha,\beta,\gamma\}: \ roots \ of \ f(x)\\ &Primitive \ element \quad v=1\cdot\alpha+2\cdot \beta+3\cdot\gamma \end{align*}

\[\qquad The \ system \ of \ equations\]

\[ \qquad \left\{ \begin{array}{l} \alpha^3-3\alpha+1=0\\ \beta^2+\alpha\beta+\alpha^2-3=0\\ \alpha+\beta+\gamma=0\\ v-(\alpha+2\beta+3\gamma)=0 \end{array} \right.\\ \qquad \qquad \qquad \Downarrow \]

\[\qquad Elimination \ Theory\]

\[ \qquad V(v)= v^6-18v^4+81v^2-81 \\ \qquad \qquad =\left( {{v}^{3}}-9 v-9\right) \, \left( {{v}^{3}}-9 v+9\right) \]

\[g_{0}(x)=x^3-9x-9 \]

\[ g_0(x):minimal \ polynomial \ of \ v \ on \ F_0=Q(\omega) \]

\[Factorization \ of \ f(x) \ on \ F_0(v)\] \[\quad "maxima's \ function \ "\] \[\qquad factor(f(x),g_0(v))\]

\begin{align*} \alpha&=-\frac{{{v}^{2}}}{3}+2 & \beta&=\frac{2 {{v}^{2}}}{3}-v-4\\ \gamma&=-\frac{{{v}^{2}}}{3}+v+2 & & \end{align*}

\begin{align*} v_{1}&=v & v_{2}&={{v}^{2}}-v-6\\ v_{3}&=-{{v}^{2}}+2 v+6 & v_{4}&=-{{v}^{2}}+v+6\\ v_{5}&={{v}^{2}}-2 v-6 & v_{6}&=-v \end{align*}

\begin{align*} &g_0(v_i)=0 \quad for \ (i=1,4,5) \\ &\qquad \qquad \Downarrow\ \\ &A_3: Galois \ group \ of \ F(x) \\ &\qquad composition \ series \quad A_3 \rhd \{e\} \end{align*}

\begin{align*} &g_1(x) : \ minimal \ polynomial \ of \ v\\ &g_{1}(x)=x-\frac{{{a}_{1}^{2}} \omega }{3}-\frac{2 {{a}_{1}^{2}}}{3}+{a_1} \in F_{1}[x] \\ \end{align*}

\begin{align*} \\ F_1=F_0(a_1) \quad Here \ B_1&={{a}_{1}^{3}}-3\omega +3 =0, \\ \Omega&=\omega^2+\omega+1=0 \end{align*}

\begin{align*} v=&\frac{{{a}_{1}^{2}} \omega }{3}+\frac{2 {{a}_{1}^{2}}}{3}-{a_1} \\ \\ \alpha=&\frac{{a_1} \omega }{3}-\frac{{{a}_{1}^{2}}}{3}+\frac{2 {a_1}}{3}\\ \beta=&-\frac{{{a}_{1}^{2}} \omega }{3}-\frac{2 {a_1} \omega }{3}-\frac{{a_1}}{3}\\ \gamma=&\frac{{{a}_{1}^{2}} \omega }{3}+\frac{{a_1} \omega }{3}+\frac{{{a}_{1}^{2}}}{3}-\frac{{a_1}}{3}\\ \\ Here &\quad B_1={{a}_{1}^{3}}-3\omega +3 =0,\\ &\quad \Omega=\omega^2+\omega+1=0 \end{align*}