Techniques of Solving Equations à la Galois
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[3-10] Computation for \(F_2/F_1\): finding the minimal polynomial \(g_2(x)\) (1)
The computations in this section correspond to the green part of Fig. 3-3.
The Galois group applied here is the quotient group \(A_4/V_4\), which is the cyclic group of order 3, \(C_3\). Writing \(C_3\cong \{\kappa_1,\kappa_2,\kappa_3\}\), each \(\kappa_i\) is a coset and hence consists of 4 elements of \(S_4\).
| \( i \backslash j \) | \(\kappa_1\) | \(\kappa_2\) | \(\kappa_3\) |
|---|---|---|---|
| \(\kappa_1\) | \(\kappa_1\) | \(\kappa_2\) | \(\kappa_3\) |
| \(\kappa_2\) | \(\kappa_2\) | \(\kappa_3\) | \(\kappa_1\) |
| \(\kappa_3\) | \(\kappa_3\) | \(\kappa_1\) | \(\kappa_2\) |
\begin{align} &A_4/V_4 \cong C_3 \equiv \{\kappa_1,\kappa_2,\kappa_3\} \\ \notag \\ &\left\{ \begin{array}{l} \kappa_1\equiv \{\rho_{1},\rho_{8},\rho_{17},\rho_{24}\} \\ \kappa_2\equiv \{\rho_{4},\rho_{12},\rho_{13},\rho_{21}\} \\ \kappa_3\equiv \{\rho_{5},\rho_{9},\rho_{16},\rho_{20}\} \\ \end{array} \right.\\ \end{align}
[Step 1] LRT (Lagrange Resolvent Transformation)
\begin{align}
&h_0=\prod_{\rho_i \in \ \kappa_1}\rho_i(x-v)=(x-v_1)(x-v_8)(x-v_{17})(x-v_{24}) \notag \\
&h_1=\prod_{\rho_i \in \ \kappa_2}\rho_i(x-v)=(x-v_4)(x-v_{12})(x-v_{13})(x-v_{21}) \\
&h_2=\prod_{\rho_i \in \ \kappa_3}\rho_i(x-v)=(x-v_5)(x-v_9)(x-v_{16})(x-v_{20}) \notag \\
\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}
( \ t_1: \ Lagrange \ resolvent \ )\\
\\
\Omega=\omega^2+\omega+1=0 \\
\end{array}\\
\end{align}
With this setup, we examine the polynomials \(\{h_0,h_1,h_2\}\) in \(x\) defined in (10.3) from the viewpoint of automorphisms.
As shown in (10.5), write the coefficients of \(\{h_0,h_1,h_2\}\) as \(\{ \ ca_i,cb_i,cc_i \ [i=0,1,2,3] \ \}\).
The coefficients \(\{ca_i,cb_i,cc_i\}\) are composed of the \(v_i\) belonging to \(\{v \kappa_1,v\kappa_2,v\kappa_3\}\), respectively, as in (10.6).
\begin{align} &\left\{ \begin{array}{l} h_0=x^4+ca_3x^3+ca_2x^2+ca_1x+ca_0 \\ h_1=x^4+cb_3x^3+cb_2x^2+cb_1x+cb_0\\ h_2=x^4+cc_3x^3+cc_2x^2+cc_1x+cc_0 \qquad \{ h_0,h_1,h_2 \}\ \in F_1(v)[x]\\ \end{array} \right.\\ \notag \\ &\quad ca_3=-(v_{1}+v_{8}+v_{17}+v_{24}), \quad cb_3=-(v_{4}+v_{12}+v_{13}+v_{21}), \notag \\ & \quad cc_3=-(v_{5}+v_{9}+v_{16}+v_{20}), \quad etc. \qquad \{ ca_i,cb_i,cc_i \} \ \in F_1(v) \\ \end{align}
| \( \ \) | \(\kappa_i(v \kappa_1)\) | \(\kappa_i(v \kappa_2)\) | \(\kappa_i(v \kappa_3)\) |
|---|---|---|---|
| \(\kappa_1\) | \(v_{1},v_{8},v_{17},v_{24}\) | \(v_{4},v_{12},v_{13},v_{21}\) | \(v_{5},v_{9},v_{16},v_{20}\) |
| \(\kappa_2\) | \(v_{4},v_{12},v_{13},v_{21}\) | \(v_{5},v_{9},v_{16},v_{20}\) | \(v_{1},v_{8},v_{17},v_{24}\) |
| \(\kappa_3\) | \(v_{5},v_{9},v_{16},v_{20}\) | \(v_{1},v_{8},v_{17},v_{24}\) | \(v_{4},v_{12},v_{13},v_{21}\) |
From [Table 3-9] we see that the three groups \(\{v \kappa_1,v\kappa_2,v\kappa_3\}\) of the \(v_i\) do not mix under the automorphisms \(\{\kappa_1,\kappa_2,\kappa_3\}\). Therefore the coefficients \(\{ca_i,cb_i,cc_i\}\) of the polynomials \(h_0,h_1,h_2\) transform under \(\{\kappa_1,\kappa_2,\kappa_3\}\) as in Table 3-10. Likewise, the images of \(\{h_0,h_1,h_2\}\) under \(\{\kappa_1,\kappa_2,\kappa_3\}\) are summarized in Table 3-11.
| \( \ \) | \(\kappa_j(ca_i)\) | \(\kappa_j(cb_i)\) | \(\kappa_j(cc_i)\) |
|---|---|---|---|
| \(\kappa_1\) | \(ca_i\) | \(cb_i\) | \(cc_i\) |
| \(\kappa_2\) | \(cb_i\) | \(cc_i\) | \(ca_i\) |
| \(\kappa_3\) | \(cc_i\) | \(ca_i\) | \(cb_i\) |
\(\quad \Rightarrow \quad \)
| \( \ \) | \(\kappa_j(h_0)\) | \(\kappa_j(h_1)\) | \(\kappa_j(h_2)\) |
|---|---|---|---|
| \(\kappa_1\) | \(h_0\) | \(h_1\) | \(h_2\) |
| \(\kappa_2\) | \(h_1\) | \(h_2\) | \(h_0\) |
| \(\kappa_3\) | \(h_2\) | \(h_0\) | \(h_1\) |
Next, the coefficients of the polynomials \(\{t_0,t_1,t_2\}\) produced by the Lagrange Resolvent Transformation in (10.4) are combinations of the coefficients \(\{ca_i,cb_i,cc_i\}\) as indicated in (10.7). Hence the images of \(\{t_0,t_1,t_2\}\) under \(\{\kappa_1,\kappa_2,\kappa_3\}\) are summarized in [Table 3-12].
\begin{align} &\left\{ \begin{array}{l} t_0=\frac{1}{3}(h_0+h_1+h_2)=x^{4}+\displaystyle\sum_{i=0}^{3} \frac{1}{3} (ca_i+cb_i+cc_i)x^i=x^{4}+\displaystyle\sum_{i=0}^{3} cd_i x^i \\ t_1=\frac{1}{3}(h_0+\omega h_1+\omega^2 h_2)=\displaystyle \sum_{i=0}^{3} \frac{1}{3} (ca_i+\omega cb_i+\omega^2 cc_i)x^i=\displaystyle \sum_{i=0}^{3} ce_i x^i \\ t_2=\frac{1}{3}(h_0+\omega^2 h_1+\omega h_2)=\displaystyle \sum_{i=0}^{3} \frac{1}{3} (ca_i+\omega^2 cb_i+\omega cc_i)x^i=\displaystyle \sum_{i=0}^{3} ck_i x^i \\ \end{array} \right.\\ \end{align}
| \( \ \) | \(\kappa_j(cd_i)\) | \(\kappa_j(ce_i)\) | \(\kappa_j(ck_i)\) | \(\kappa_j(t_0)\) | \(\kappa_j(ce_i^3)\) | \(\kappa_j(ck_i^3)\) | \(\kappa_j(ce_i \cdot ck_l)\) |
|---|---|---|---|---|---|---|---|
| \(\kappa_1\) | \(cd_i\) | \(ce_i\) | \(ck_i\) | \(t_0\) | \(ce_i^3\) | \(ck_i^3\) | \(ce_i \cdot ck_l\) |
| \(\kappa_2\) | \(cd_i\) | \(\omega^2 \cdot ce_i\) | \(\omega \cdot ck_i\) | \(t_0\) | \(ce_i^3\) | \(ck_i^3\) | \(ce_i \cdot ck_l\) |
| \(\kappa_3\) | \(cd_i\) | \(\omega \cdot ce_i\) | \(\omega^2 \cdot ck_i\) | \(t_0\) | \(ce_i^3\) | \(ck_i^3\) | \(ce_i \cdot ck_l\) |
From [Table 3-12] we see that \(t_0\) is invariant under all permutations by \(\{\kappa_1,\kappa_2,\kappa_3\}\). Hence \(t_0 \in F_1[x]\). The yellow-highlighted entries in [Table 3-12] show that \(\{ce_i^3, ck_i^3, ce_i \cdot ck_l\}\) are also invariant under \(\{\kappa_1,\kappa_2,\kappa_3\}\). Therefore \(\{ce_i^3, ck_i^3, ce_i \cdot ck_l\} \in F_1\) as well.
\begin{align} t_0 \ \in F_1[x], \qquad \{t_1,t_2\} \ \in F_1(v)[x], \qquad \{ce_i^3,ck_i^3,ce_i \cdot ck_l\} \ \in F_1 \\ \end{align}
[Supplement] Perhaps unnecessary, but here is how to obtain part of [Table 3-12], illustrated with the 3rd and 4th rows of the 3rd column.
\begin{align} \kappa_2(ce_i)&=\frac{1}{3} \kappa_2(ca_i+\omega cb_i+ \omega^2 cc_i)=\frac{1}{3} \biggl[ \kappa_2(ca_i)+\omega \kappa_2(cb_i)+ \omega^2 \kappa_2(cc_i)\frac{1}{3} \biggr] \notag \\ &=\frac{1}{3} ( cb_i+\omega cc_i +\omega^2 ca_i )= \frac{\omega^2}{3} (ca_i+\omega cb_i +\omega^2 cc_i)=\omega^2 ce_i \notag \\ \kappa_3(ce_i)&=\frac{1}{3} \kappa_3(ca_i+\omega cb_i+ \omega^2 cc_i)=\frac{1}{3} \biggl[\kappa_3(ca_i)+\omega \kappa_3(cb_i)+ \omega^2 \kappa_3(cc_i) \frac{1}{3} \biggr] \notag \\ &=\frac{1}{3} ( cc_i+\omega ca_i +\omega^2 cb_i )= \frac{\omega}{3} (ca_i+\omega cb_i +\omega^2 cc_i)=\omega ce_i \notag \\ \end{align}