Techniques for Solving Solvable Equations via Galois Theory

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[3-8] Computation for \(F_1/F_0\): finding the minimal polynomial \(g_1(x)\) (1)

We have outlined the overall flow in the previous section, so we now proceed to compute \(g_1(x)\). The calculation corresponds to the green block in Fig. 3-2 below.

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The computation to obtain the minimal polynomial \(g_1(x)\) of \(v\) is divided into three steps. We begin with the first step.

[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_4)...(x-v_{21})(x-v_{24}) \\ & h_1=\prod_{\rho_i \in \ \kappa_2}\rho_i(x-v)=(x-v_2)(x-v_3)...(x-v_{22})(x-v_{23}) \notag \\ \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} \\ \end{align}

The Galois group applied here is the quotient \(S_4/A_4 \cong C_2\), i.e., the cyclic group \(C_2\) of order \(2\). Writing its elements as \(\{\kappa_1,\kappa_2\}\), each is a coset and thus consists of 12 elements.
We also classify the \(v_i\) into two groups \(\{v \kappa_1, v\kappa_2\}\) corresponding to \(\{\kappa_1,\kappa_2\}\).

\begin{align} &S_4/A_4 \cong C_2 \equiv \{\kappa_1,\kappa_2\} \quad \kappa_2^2=\kappa_1 \\ \notag \\ &\left\{ \begin{array}{l} \kappa_1\equiv \{\rho_{1},\rho_{4},\rho_{5},\rho_{8},\rho_{9},\rho_{12},\rho_{13},\rho_{16},\rho_{17},\rho_{20},\rho_{21},\rho_{24}\} \\ \kappa_2\equiv \{\rho_{2},\rho_{3},\rho_{6},\rho_{7},\rho_{10},\rho_{11},\rho_{14},\rho_{15},\rho_{18},\rho_{19},\rho_{22},\rho_{23}\} \\ \end{array} \right.\\ \notag \\ &\left\{ \begin{array}{l} v \kappa_1\equiv \{v_{1},v_{4},v_{5},v_{8},v_{9},v_{12},v_{13},v_{16},v_{17},v_{20},v_{21},v_{24}\} \\ v \kappa_2\equiv \{v_{2},v_{3},v_{6},v_{7},v_{10},v_{11},v_{14},v_{15},v_{18},v_{19},v_{22},v_{23}\} \\ \end{array} \right.\\ \end{align}

With this preparation, we examine the polynomials \(\{h_0,h_1\}\) in \(x\), defined in (8.1), from the viewpoint of the group action.
As in (8.6), write the coefficients of \(\{h_0,h_1\}\) as \(\{ca_i,cb_i\,[i=0,1,\ldots,11]\}\). The coefficients \(\{ca_i,cb_i\}\) are composed of the \(v_i\) belonging to \(\{v \kappa_1, v\kappa_2\}\), respectively, as in (8.7).

\begin{align} &\left\{ \begin{array}{l} h_0=x^{12}+ca_{11}x^{11}+...+ca_1x+ca_0 \\ \\ h_1=x^{12}+cb_{11}x^{11}+...+cb_1x+cb_0 \\ \end{array} \right.\\ \notag \\ &\quad ca_{11}=-(v_1+v_4+...+v_{24}), \quad cb_{11}=-(v_2+v_3+....+v_{23}), \quad \text{etc.} \\ \end{align}

[Table 3-3] Action \(\kappa_j(v\kappa_i)\)
\( \ \)	\(\kappa_j(v \kappa_1)\)	\(\kappa_j(v \kappa_2)\)
\(\kappa_1\)	\(v_{1},v_{4},v_{5},v_{8},v_{9},v_{12}\) \(v_{13},v_{16},v_{17},v_{20},v_{21},v_{24}\)	\(v_{2},v_{3},v_{6},v_{7},v_{10},v_{11}\) \(v_{14},v_{15},v_{18},v_{19},v_{22},v_{23}\)
\(\kappa_2\)	\(v_{2},v_{3},v_{6},v_{7},v_{10},v_{11}\) \(v_{14},v_{15},v_{18},v_{19},v_{22},v_{23}\)	\(v_{1},v_{4},v_{5},v_{8},v_{9},v_{12}\) \(v_{13},v_{16},v_{17},v_{20},v_{21},v_{24}\)

From [Table 3-3] we see that the two groups of \(v_i\), \(\{v \kappa_1, v\kappa_2\}\), are not mixed by the automorphisms \(\{\kappa_1,\kappa_2\}\). Therefore, under the action of \(\{\kappa_1,\kappa_2\}\), the coefficients \(\{ca_i,cb_i\}\) of the polynomials \(h_0,h_1\) transform as in [Table 3-4]. Similarly, the images of \(\{h_0,h_1\}\) under \(\{\kappa_1,\kappa_2\}\) are as in [Table 3-5].

[Table 3-4] Action \(\kappa_j(ca_i,cb_i)\)
\( \ \)	\(\kappa_j(ca_i)\)	\(\kappa_j(cb_i)\)
\(\kappa_1\)	\(ca_i\)	\(cb_i\)
\(\kappa_2\)	\(cb_i\)	\(ca_i\)

\(\quad \Rightarrow \quad \)

[Table 3-5] Action \(\kappa_j(h_i)\)
\( \ \)	\(\kappa_j(h_0)\)	\(\kappa_j(h_1)\)
\(\kappa_1\)	\(h_0\)	\(h_1\)
\(\kappa_2\)	\(h_1\)	\(h_0\)

Next, by (8.2), the coefficients of the polynomials \(\{t_0,t_1\}\) generated by the Lagrange Resolvent Transformation are combinations of the coefficients \(\{ca_i,cb_i\}\) as shown in (8.8). Consequently, their images under \(\{\kappa_1,\kappa_2\}\) are given in [Table 3-7].

\begin{align} &\left\{ \begin{array}{l} t_0=\frac{1}{2}(h_0+h_1)=x^{12}+\displaystyle\sum_{i=0}^{11} \frac{1}{2}(ca_i+cb_i)x^i=x^{12}+\displaystyle\sum_{i=0}^{11} cc_i x^i\\ t_1=\frac{1}{2}(h_0-h_1)=\displaystyle \sum_{i=0}^{11}\frac{1}{2} (ca_i-cb_i)x^i= \displaystyle \sum_{i=0}^{11} cd_i x^i\\ \end{array} \right.\\ \end{align}

[Table 3-6] Action \(\kappa_j(ca_i \pm cb_i)\)
\( \ \)	\(\kappa_j(cc_i=\frac{1}{2}(ca_i+cb_i))\)	\(\kappa_j(cd_i=\frac{1}{2}(ca_i-cb_i))\)
\(\kappa_1\)	\(cc_i=\frac{1}{2}(ca_i+cb_i)\)	\(cd_i=\frac{1}{2}(ca_i-cb_i)\)
\(\kappa_2\)	\(cc_i=\frac{1}{2}(cb_i+ca_i)\)	\(-cd_i=\frac{1}{2}(cb_i-ca_i)\)

\(\quad \Rightarrow \quad \)

[Table 3-7] Action \(\kappa_j(t_i)\)
\( \ \)	\(\kappa_j(t_0)\)	\(\kappa_j(t_1)\)	\(\kappa_j(cd_i ^2)\)
\(\kappa_1\)	\(t_0\)	\(t_1\)	\(cd_i^2\)
\(\kappa_2\)	\(t_0\)	\(-t_1\)	\(cd_i^2\)

From [Table 3-7], \(t_0\) is invariant under the permutation action of all elements of \(\{\kappa_1,\kappa_2\}\). Hence \(t_0 \in F_0[x]\). Also, as highlighted in yellow in [Table 3-7], each \(cd_i^2\) is invariant under the action of \(\{\kappa_1,\kappa_2\}\), so \(cd_i^2 \in F_0\). Summarizing:

\begin{align} t_0 \ \in F_0[x], \quad t_1 \ \in F_0(v)[x], \quad cd_i^2 \ \in F_0 \\ \end{align}

Let us look a little more closely at \(t_1\). Since the coefficients of the polynomial \(t_1\) in \(x\) are composed of the \(v_i\), they lie in \(F_0(v)\). Let \(m\) be the highest power of \(x\) appearing with nonzero coefficient in \(t_1\), and write the coefficient of \(x^m\) as \((ca_m-cb_m)=cd_m \neq 0\). Factoring out \(cd_m\) from \(t_1\) yields (8.10). We then ask: in which field does the monic polynomial \(q_1(x)\) lie?

\begin{align} &t_1=cd_m \times \bbox[#FFFF00]{ \biggl( x^m+\displaystyle \sum_{i=0}^{m-1} \frac{cd_i}{cd_m}x^i\biggr) }=cd_m \times \bbox[#FFFF00]{ q_1(x) }\\ \notag \\ &\kappa_1 \biggl( \frac{cd_i}{cd_m}\biggr)= \frac{cd_i}{cd_m} \quad [i=0,1,...,m-1] \notag \\ &\kappa_2 \biggl( \frac{cd_i}{cd_m}\biggr)= \frac{-cd_i}{-cd_m}=\frac{cd_i}{cd_m} \quad [i=0,1,...,m-1] \notag \\ \notag \\ &\therefore \ \biggl( \frac{cd_i}{cd_m}\biggr) \ \in F_0 \quad \Rightarrow \quad \bbox[#FFFF00]{ q_1(x) \in F_0[x] } \\ \end{align}

In conclusion, every coefficient of \(q_1(x)\) is invariant under the action of \(\{\kappa_1,\kappa_2\}\). Therefore \(q_1(x)\) is a polynomial over \(F_0\). All of the above follows purely from the behavior of the \(v_i\) under the automorphisms \(\rho_i\). In short, we have seen one aspect of how quotient groups and automorphisms of extension fields intertwine to drive the theory forward.

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