\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*}
\begin{align} \setCounter{37} \beta&=\frac{2v^2}{3}-v-4, \qquad v_4=-v^2+v+6 \notag \\ \therefore \ \sigma_4(\beta)&=\sigma_4\left(\frac{2v^2}{3}-v-4 \right)=\frac{2v_4^2}{3}-v_4-4 \\ &=\frac{2(-v^2+v+6)^2}{3}-(-v^2+v+6)-4 \notag \\ &=\frac{2 {{v}^{4}}}{3}-\frac{4 {{v}^{3}}}{3}-\frac{19 {{v}^{2}}}{3}+7 v+14\notag \\ &=-\frac{{{v}^{2}}}{3}+v+2=\gamma \quad ( \ mod \ g_0(v) \ ) \\ \therefore \ \sigma_4(\beta)&=\gamma\\ \end{align}
\( i \backslash j \) | \(\sigma_i(\alpha)\) | \(\sigma_i(\beta)\) | \(\sigma_i(\gamma)\) |
---|---|---|---|
\(\sigma_1\) | \(\alpha\) | \(\beta\) | \(\gamma\) |
\(\sigma_4\) | \(\beta\) | \(\gamma\) | \(\alpha\) |
\(\sigma_5\) | \(\gamma\) | \(\alpha\) | \(\beta\) |
\begin{align} Gal(F_0(v)/F_0): \ Galois \ group \ of \ f(x) =\{\sigma_1,\sigma_4,\sigma_5\}=A_3 \\ \end{align}
\begin{align} &【例題1】\quad F_0(v) \quad \Rightarrow \quad F_0(v,a_1) \quad \Rightarrow \quad F_0(a_1,a_2) \notag \\ &\qquad [\alpha(v),\beta(v),\gamma(v)] \qquad \Longrightarrow \qquad [\alpha(a_1,a_2),\beta(a_1,a_2),\gamma(a_1,a_2)]\notag \\ \notag \\ &【例題2】\quad F_0(v) \qquad \quad \Rightarrow \qquad F_0(a_1)\notag \\ &\qquad [\alpha(v),\beta(v),\gamma(v)] \quad \Rightarrow \quad [\alpha(a_1),\beta(a_1),\gamma(a_1)]\notag \\ \end{align}
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1st upload: 2023/06/17
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