Techniques of Solving Equations à la Galois
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1st upload: 2023/06/17
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[2-8] The Galois group of the equation \(f(x)\) and its composition series
From the discussion in the previous section, we found that the set of all \(F_0\)-automorphisms \(\{\rho_{1}, \rho_{2},..., \rho_{6}\}\) has the structure of the symmetric group \(S_3\).Assuming some group theory: the symmetric group \(S_3\) has a composition series as shown in (8.1). In Galois theory, as indicated in (8.1)–(8.2), there are three fields corresponding to the three groups in this composition series.
Moreover, the field extensions \(\{F_1/F_0,\ F_2/F_1\}\) are Galois extensions; their extension degrees are prime and, as shown in (8.3)–(8.4), each is a cyclic extension of degrees 2 and 3, respectively.
The process of enlarging the base field \(F_0\) step by step to \(F_1\) and \(F_2\) constitutes the method of solving the equation using Galois theory.
\begin{align} &Gal(F_0(v)/F_0) =\{\rho_{1}, \rho_{2},..., \rho_{6}\} =S_3 : \ Galois \ group \ of \ F_0(v)/F_0 \notag \\ \notag \\ & \ Composition \ series \ of \ Galois \ group \ S_3 \notag \\ \end{align} \begin{align} &S_3 & &\rhd & &A_3 & &\rhd & &e \\ &\updownarrow & & & &\updownarrow & & & &\updownarrow \notag \\ &F_0 & &\rightarrow & &F_1 & &\rightarrow & &F_2 \ \ ( \cong F_0(v) ) \\ \end{align} \begin{align} & \qquad \qquad \Downarrow \notag \\ \notag \\ & Galois \ extension & & & &Galois \ Group \notag \\ \notag \\ &[1] \quad [ \ F_1:F_0 \ ]=2 & &\rightarrow & &Gal(F_1/F_0) = S_3/A_3 \cong C_2 \\ \notag \\ &[2] \quad [ \ F_2:F_1 \ ]=3 & &\rightarrow & &Gal(F_2/F_1) = A_3/e \cong C_3\\ \end{align}
(Supplement) The elements of the Galois groups that correspond to the above composition series are as follows.
\begin{align} &S_3=\{\rho_{1},\rho_{2},\rho_{3},\rho_{4},\rho_{5},\rho_{6}\}, \quad A_3=\{\rho_{1},\rho_{4},\rho_{5}\}, \quad e=\{\rho_{1}\} \notag \\ \notag \\ &\left\{ \begin{array}{l} S_3/A_3 \cong C_2 \equiv \{\kappa_1,\kappa_2\}, \quad \kappa_1=\{\rho_{1},\rho_{4},\rho_{5}\} ,\quad \kappa_2=\{\rho_{2},\rho_{3},\rho_{6}\}\\ A_3/e \cong C_3=\{\rho_{1},\rho_{4},\rho_{5}\}\\ \end{array} \right. \notag \\ \end{align}
[2-9] Computing \(F_1/F_0\): finding the minimal polynomial \(g_1(x)\) (1)
The light-blue part of (Fig. 2-2) (the computation of \(\omega\)) was already explained in Chapter 1.
In this section we compute the minimal polynomial \(g_1(x)\). The calculation has three stages.
First, we carry out Stage 1 in the boxed display below.
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_5) \\
&h_1=\prod_{\rho_i \in \ \kappa_2}\rho_i(x-v)=(x-v_2)(x-v_3)(x-v_6) \\
\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}
Let us see how \(\{h_0,h_1\}\) transform under the elements \(\{\kappa_1,\kappa_2\}\) of \(Gal(F_1/F_0)\). We first expand \(\{h_0,h_1\}\).
\begin{align} h_0&=(x-v_1)(x-v_4)(x-v_5) & h_1&=(x-v_2)(x-v_3)(x-v_6) \notag \\ &=x^3+ca_2x^2+ca_1x+ca_0 & &=x^3+cb_2x^2+cb_1x+cb_0\\ \notag \\ &\left\{ \begin{array}{l} ca_2=-({v_5}+{v_4}+{v_1})\\ ca_1={v_4} {v_5}+{v_1} {v_5}+{v_1} {v_4}\\ ca_0=-{v_1} {v_4} {v_5}\\ \end{array} \right. & &\left\{ \begin{array}{l} cb_2=-({v_6}+{v_3}+{v_2})\\ cb_1={v_3} {v_6}+{v_2} {v_6}+{v_2} {v_3}\\ cb_0=-{v_2} {v_3} {v_6}\\ \end{array} \right.\\ \end{align}
| \(\rho_i(v_1)\) | \(\rho_i(v_4)\) | \(\rho_i(v_5)\) | \(\rho_i(v_2)\) | \(\rho_i(v_3)\) | \(\rho_i(v_6)\) | ||
| \(\kappa_1\) | \(\rho_1\) | \(v_1\) | \(v_4\) | \(v_5\) | \(v_2\) | \(v_3\) | \(v_6\) |
| \(\rho_4\) | \(v_4\) | \(v_5\) | \(v_1\) | \(v_3\) | \(v_6\) | \(v_2\) | |
| \(\rho_5\) | \(v_5\) | \(v_1\) | \(v_4\) | \(v_6\) | \(v_2\) | \(v_3\) | |
| \(\kappa_2\) | \(\rho_2\) | \(v_2\) | \(v_6\) | \(v_3\) | \(v_1\) | \(v_5\) | \(v_4\) |
| \(\rho_3\) | \(v_3\) | \(v_2\) | \(v_6\) | \(v_4\) | \(v_1\) | \(v_5\) | |
| \(\rho_6\) | \(v_6\) | \(v_3\) | \(v_2\) | \(v_5\) | \(v_4\) | \(v_1\) |
(No further explanation should be necessary.)
[Table 2-4]
\(\quad \Downarrow\)
[Table 2-5]
\(\quad \Downarrow\)
[Table 2-6]
| \( \ \) | \(\kappa_i(ca_i)\) | \(\kappa_i(cb_j)\) |
|---|---|---|
| \(\kappa_1\) | \(ca_i\) | \(cb_j\) |
| \(\kappa_2\) | \(cb_i\) | \(ca_j\) |
\( \ \Rightarrow \ \)
| \( \ \) | \(\kappa_i(h_0)\) | \(\kappa_i(h_1)\) |
|---|---|---|
| \(\kappa_1\) | \(h_0\) | \(h_1\) |
| \(\kappa_2\) | \(h_1\) | \(h_0\) |
From [Table 2-6] we see that \(\{h_0,h_1\}\) are invariant under the action of \(\kappa_1\), whereas under \(\kappa_2\) they are interchanged. Because \(\{h_0,h_1\}\) have this property, it is easy to infer that the \(\{t_0,t_1\}\) produced by the Lagrange resolvent transformation (9.3) have the properties listed in [Table 2-8] below.
| \( \ \) | \(\kappa_i(h_0+h_1)\) | \(\kappa_i(h_0-h_1)\) |
|---|---|---|
| \(\kappa_1\) | \(h_0+h_1\) | \(h_0-h_1\) |
| \(\kappa_2\) | \(h_1+h_0\) | \(h_1-h_0\) |
\( \ \Rightarrow \ \)
| \( \ \) | \(\kappa_i(t_0)\) | \(\kappa_i(t_1)\) | \(\kappa_i(t_1^2)\) |
|---|---|---|---|
| \(\kappa_1\) | \(t_0\) | \(t_1\) | \(t_1^2\) |
| \(\kappa_2\) | \(t_0\) | \(-t_1\) | \(t_1^2\) |
All of the above followed solely from the mapping behavior of the automorphisms \(\rho_i\) on the \(v_i\). It is interesting how many properties can be read off from the mapping behavior alone, even without explicit computation. This gives a good sense of the strength of automorphisms, which frequently appear in Galois theory. The summary of this section is as follows.
\begin{align} &t_0 \ \in \ F_0[x], \quad t_1 \ \notin \ F_0[x] \\ \notag \\ &\kappa_2(t_1^2)=\kappa_2(t_1) \cdot \kappa_2(t_1)=(-t_1)^2=t_1^2 \qquad \therefore \ t_1^2 \ \in \ F_0[x] \\ \end{align}