Techniques for Solving Solvable Equations via Galois Theory

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[3-5] Verifying that the maps \(\rho_i\) carry a group structure

Since we have established that \(F_0(v)\) is a Galois extension of \(F_0\), the following theorem of Galois theory applies.

“There exist as many field automorphisms of \(F_0(v)\) as there are conjugate roots of \(g_0(x)\), each sending the primitive element \(v\) to a conjugate root; moreover, these automorphisms form a group.”

In this section we verify that this theorem holds here.
To that end, we take \(\{\rho_i\}\ [i=1,2,\ldots,24]\) defined in (5.1) as candidates for such automorphisms. Equation (5.1) regards the polynomial expressions of \(v_i\) from (4.10–13) as the images of \(v\) under \(\rho_i\). We then check that the \(\rho_i\) form a group isomorphic to the symmetric group \(S_4\), and that each \(\rho_i\) is an automorphism mapping conjugate roots to conjugate roots.

\begin{align} \rho_{1}(v) &\equiv v_1=v \notag \\ \notag \\ \rho_{2}(v) &\equiv v_2=\frac{801167701943012874015343807}{2531636596706154203109159208286706204672}v^{23}-\frac{51207699710669004924125}{99737485589022345786910893443907584}v^{22} \notag \\ &\quad .......... \notag \\ &+\frac{4663968749846627680850109200993107}{2971516062464968406218495540075104}v+\frac{2718803338720300088760700554765}{1521873254227025539961408896544} \\ \notag \\ &\qquad \qquad .........\notag \\ \notag \\ \rho_{23}(v) &\equiv v_{23}=-\frac{801167701943012874015343807}{2531636596706154203109159208286706204672}v^{23}-\frac{51207699710669004924125}{99737485589022345786910893443907584}v^{22} \notag \\ &\quad .......... \notag \\ &-\frac{4663968749846627680850109200993107}{2971516062464968406218495540075104}v+\frac{2718803338720300088760700554765}{1521873254227025539961408896544} \notag \\ \notag \\ \rho_{24}(v) &\equiv v_{24}=-v \notag \\ \end{align}

We must confirm that \(\rho_i(v)\), \(i=1,2,\ldots,24\), form a group under composition. For this, we compile the multiplication table \(\rho_i \circ \rho_j = \rho_k\) and check whether the compositions yield a group. The verification procedure is exactly the same as in Chapter 2, Section [2-6]. In Chapter 3 the formulas for \(v_i\) are too long to present the intermediate computations, so we simply display the resulting table below .

From this table we find that the maps \(\rho_i\) form a group under composition that is equivalent to the symmetric group \(S_4\).

[Table 3-1] Multiplication table for \(S_4\): \(\rho_i \circ \rho_j =\rho_k\) (entries shown as numbers only)
\( i \backslash j \) \(1 \) \(2 \) \(3 \) \(4 \) \(5 \) \(6 \) \(7 \) \(8 \) \(9 \) \({10} \) \({11} \) \({12} \) \({13} \) \({14} \) \({15} \) \({16} \) \({17} \) \({18} \) \({19} \) \({20} \) \({21} \) \({22} \) \({23} \) \({24}\)

\(1\) \(1 \) \(2 \) \(3 \) \(4 \) \(5 \) \(6 \) \(7 \) \(8 \) \(9 \) \({10} \) \({11} \) \({12} \) \({13} \) \({14} \) \({15} \) \({16} \) \({17} \) \({18} \) \({19} \) \({20} \) \({21} \) \({22} \) \({23} \) \({24}\)

\(2\) \(2 \) \(1 \) \(5 \) \(6 \) \(3 \) \(4 \) \(8 \) \(7 \) \({11} \) \({12} \) \(9 \) \({10} \) \({19} \) \({20} \) \({21} \) \({22} \) \({23} \) \({24}\) \({13} \) \({14} \) \({15} \) \({16} \) \({17} \) \({18}\)

\(3 \) \(3 \) \(4 \) \(1 \) \(2 \) \(6 \) \(5 \) \({13} \) \({14} \) \({15} \) \({16} \) \({17} \) \({18} \) \(7 \) \(8 \) \(9 \) \({10} \) \({11} \) \({12} \) \({20} \) \({19} \) \({23} \) \({24}\) \({21} \) \({22} \)

\(4\) \(4 \) \(3 \) \(6 \) \(5 \) \(1 \) \(2 \) \({14} \) \({13} \) \({17} \) \({18} \) \({15} \) \({16} \) \({20} \) \({19} \) \({23} \) \({24}\) \({21} \) \({22} \) \(7 \) \(8 \) \(9 \) \({10} \) \({11} \) \({12} \)

\(5 \) \(5 \) \(6 \) \(2 \) \(1 \) \(4 \) \(3 \) \({19} \) \({20} \) \({21} \) \({22} \) \({23} \) \({24}\) \(8 \) \(7 \) \({11} \) \({12} \) \(9 \) \({10} \) \({14} \) \({13} \) \({17} \) \({18} \) \({15} \) \({16} \)

\(6 \) \(6 \) \(5 \) \(4 \) \(3 \) \(2 \) \(1 \) \({20} \) \({19} \) \({23} \) \({24}\) \({21} \) \({22} \) \({14} \) \({13} \) \({17} \) \({18} \) \({15} \) \({16} \) \(8 \) \(7 \) \({11} \) \({12} \) \(9 \) \({10} \)

\(7 \) \(7 \) \(8 \) \(9 \) \({10} \) \({11} \) \({12} \) \(1 \) \(2 \) \(3 \) \(4 \) \(5 \) \(6 \) \({15} \) \({16} \) \({13} \) \({14} \) \({18} \) \({17} \) \({21} \) \({22} \) \({19} \) \({20} \) \({24}\) \({23} \)

\(8 \) \(8 \) \(7 \) \({11} \) \({12} \) \(9 \) \({10} \) \(2 \) \(1 \) \(5 \) \(6 \) \(3 \) \(4 \) \({21} \) \({22} \) \({19} \) \({20} \) \({24}\) \({23} \) \({15} \) \({16} \) \({13} \) \({14} \) \({18} \) \({17} \)

\(9 \) \(9 \) \({10} \) \(7 \) \(8 \) \({12} \) \({11} \) \({15} \) \({16} \) \({13} \) \({14} \) \({18} \) \({17} \) \(1 \) \(2 \) \(3 \) \(4 \) \(5 \) \(6 \) \({22} \) \({21} \) \({24}\) \({23} \) \({19} \) \({20} \)

\({10} \) \({10} \) \(9 \) \({12} \) \({11} \) \(7 \) \(8 \) \({16} \) \({15} \) \({18} \) \({17} \) \({13} \) \({14} \) \({22} \) \({21} \) \({24}\) \({23} \) \({19} \) \({20} \) \(1 \) \(2 \) \(3 \) \(4 \) \(5 \) \(6 \)

\({11} \) \({11} \) \({12} \) \(8 \) \(7 \) \({10} \) \(9 \) \({21} \) \({22} \) \({19} \) \({20} \) \({24}\) \({23} \) \(2 \) \(1 \) \(5 \) \(6 \) \(3 \) \(4 \) \({16} \) \({15} \) \({18} \) \({17} \) \({13} \) \({14} \)

\({12} \) \({12} \) \({11} \) \({10} \) \(9 \) \(8 \) \(7 \) \({22} \) \({21} \) \({24}\) \({23} \) \({19} \) \({20} \) \({16} \) \({15} \) \({18} \) \({17} \) \({13} \) \({14} \) \(2 \) \(1 \) \(5 \) \(6 \) \(3 \) \(4 \)

\({13} \) \({13} \) \({14} \) \({15} \) \({16} \) \({17} \) \({18} \) \(3 \) \(4 \) \(1 \) \(2 \) \(6 \) \(5 \) \(9 \) \({10} \) \(7 \) \(8 \) \({12} \) \({11} \) \({23} \) \({24}\) \({20} \) \({19} \) \({22} \) \({21} \)

\({14} \) \({14} \) \({13} \) \({17} \) \({18} \) \({15} \) \({16} \) \(4 \) \(3 \) \(6 \) \(5 \) \(1 \) \(2 \) \({23} \) \({24}\) \({20} \) \({19} \) \({22} \) \({21} \) \(9 \) \({10} \) \(7 \) \(8 \) \({12} \) \({11} \)

\({15} \) \({15} \) \({16} \) \({13} \) \({14} \) \({18} \) \({17} \) \(9 \) \({10} \) \(7 \) \(8 \) \({12} \) \({11} \) \(3 \) \(4 \) \(1 \) \(2 \) \(6 \) \(5 \) \({24}\) \({23} \) \({22} \) \({21} \) \({20} \) \({19} \)

\({16} \) \({16} \) \({15} \) \({18} \) \({17} \) \({13} \) \({14} \) \({10} \) \(9 \) \({12} \) \({11} \) \(7 \) \(8 \) \({24}\) \({23} \) \({22} \) \({21} \) \({20} \) \({19} \) \(3 \) \(4 \) \(1 \) \(2 \) \(6 \) \(5 \)

\({17} \) \({17} \) \({18} \) \({14} \) \({13} \) \({16} \) \({15} \) \({23} \) \({24}\) \({20} \) \({19} \) \({22} \) \({21} \) \(4 \) \(3 \) \(6 \) \(5 \) \(1 \) \(2 \) \({10} \) \(9 \) \({12} \) \({11} \) \(7 \) \(8 \)

\({18} \) \({18} \) \({17} \) \({16} \) \({15} \) \({14} \) \({13} \) \({24}\) \({23} \) \({22} \) \({21} \) \({20} \) \({19} \) \({10} \) \(9 \) \({12} \) \({11} \) \(7 \) \(8 \) \(4 \) \(3 \) \(6 \) \(5 \) \(1 \) \(2 \)

\({19} \) \({19} \) \({20} \) \({21} \) \({22} \) \({23} \) \({24}\) \(5 \) \(6 \) \(2 \) \(1 \) \(4 \) \(3 \) \({11} \) \({12} \) \(8 \) \(7 \) \({10} \) \(9 \) \({17} \) \({18} \) \({14} \) \({13} \) \({16} \) \({15} \)

\({20} \) \({20} \) \({19} \) \({23} \) \({24}\) \({21} \) \({22} \) \(6 \) \(5 \) \(4 \) \(3 \) \(2 \) \(1 \) \({17} \) \({18} \) \({14} \) \({13} \) \({16} \) \({15} \) \({11} \) \({12} \) \(8 \) \(7 \) \({10} \) \(9 \)

\({21} \) \({21} \) \({22} \) \({19} \) \({20} \) \({24}\) \({23} \) \({11} \) \({12} \) \(8 \) \(7 \) \({10} \) \(9 \) \(5 \) \(6 \) \(2 \) \(1 \) \(4 \) \(3 \) \({18} \) \({17} \) \({16} \) \({15} \) \({14} \) \({13} \)

\({22} \) \({22} \) \({21} \) \({24}\) \({23} \) \({19} \) \({20} \) \({12} \) \({11} \) \({10} \) \(9 \) \(8 \) \(7 \) \({18} \) \({17} \) \({16} \) \({15} \) \({14} \) \({13} \) \(5 \) \(6 \) \(2 \) \(1 \) \(4 \) \(3 \)

\({23} \) \({23} \) \({24}\) \({20} \) \({19} \) \({22} \) \({21} \) \({17} \) \({18} \) \({14} \) \({13} \) \({16} \) \({15} \) \(6 \) \(5 \) \(4 \) \(3 \) \(2 \) \(1 \) \({12} \) \({11} \) \({10} \) \(9 \) \(8 \) \(7 \)

\({24}\) \({24}\) \({23} \) \({22} \) \({21} \) \({20} \) \({19} \) \({18} \) \({17} \) \({16} \) \({15} \) \({14} \) \({13} \) \({12} \) \({11} \) \({10} \) \(9 \) \(8 \) \(7 \) \(6 \) \(5 \) \(4 \) \(3 \) \(2 \) \(1 \)

In Chapter 2 we also listed a table for \(\rho_i(v_j)\). The content is identical here, and reading it with the indices relabeled by the numbers in the table above is sufficient. In conclusion, [Table 3-1] shows that the maps \(\rho_i\) form a group equivalent to the symmetric group \(S_4\). Moreover, for every \(v_j\) we have a mapping relation of the form \(\rho_i(v_j)=v_k\).
Hence the candidates \(\{\rho_i\}\) are precisely the elements of the Galois group \(Gal(F_0(v)/F_0)\).

[3-6] The action of \(\rho_i\) on \(\{\alpha,\beta,\gamma,\delta\}\)

We now compute how \(\{ \rho_1,\rho_2,\ldots,\rho_{24}\}\) act on the four roots. Again, the expressions become unwieldy, so we omit the intermediate steps; the method is exactly the same as in Chapter 2.
[Table 3-2] records how each \(\rho_i\) acts on \([\alpha,\beta,\gamma,\delta]\).
We find that the action of \(\rho_i\) coincides exactly with the permutation action of the elements \(\sigma_i\) of \(S_4\) in (2.2) on \([\alpha,\beta,\gamma,\delta]\).

[Table 3-2] Transformation table for \(\rho_i\) (1)
\(\rho_i(\alpha)\) \(\rho_i(\beta)\) \(\rho_i(\gamma)\) \(\rho_i(\delta)\)

\(\rho_{1}\) \(\alpha\) \(\beta\) \(\gamma\) \(\delta\)

\(\rho_{2}\) \(\alpha\) \(\beta\) \(\delta\) \(\gamma\)

\(\rho_{3}\) \(\alpha\) \(\gamma\) \(\beta\) \(\delta\)

\(\rho_{4}\) \(\alpha\) \(\gamma\) \(\delta\) \(\beta\)

\(\rho_{5}\) \(\alpha\) \(\delta\) \(\beta\) \(\gamma\)

\(\rho_{6}\) \(\alpha\) \(\delta\) \(\gamma\) \(\beta\)

\(\rho_{7}\) \(\beta\) \(\alpha\) \(\gamma\) \(\delta\)

\(\rho_{8}\) \(\beta\) \(\alpha\) \(\delta\) \(\gamma\)

[Table 3-2] Transformation table for \(\rho_i\) (1)
	\(\rho_i(\alpha)\)	\(\rho_i(\beta)\)	\(\rho_i(\gamma)\)	\(\rho_i(\delta)\)
\(\rho_{1}\)	\(\alpha\)	\(\beta\)	\(\gamma\)	\(\delta\)
\(\rho_{2}\)	\(\alpha\)	\(\beta\)	\(\delta\)	\(\gamma\)
\(\rho_{3}\)	\(\alpha\)	\(\gamma\)	\(\beta\)	\(\delta\)
\(\rho_{4}\)	\(\alpha\)	\(\gamma\)	\(\delta\)	\(\beta\)
\(\rho_{5}\)	\(\alpha\)	\(\delta\)	\(\beta\)	\(\gamma\)
\(\rho_{6}\)	\(\alpha\)	\(\delta\)	\(\gamma\)	\(\beta\)
\(\rho_{7}\)	\(\beta\)	\(\alpha\)	\(\gamma\)	\(\delta\)
\(\rho_{8}\)	\(\beta\)	\(\alpha\)	\(\delta\)	\(\gamma\)

[Table 3-2] Transformation table for \(\rho_i\) (2)
\(\rho_i(\alpha)\) \(\rho_i(\beta)\) \(\rho_i(\gamma)\) \(\rho_i(\delta)\)

\(\rho_{9}\) \(\beta\) \(\gamma\) \(\alpha\) \(\delta\)

\(\rho_{10}\) \(\beta\) \(\gamma\) \(\delta\) \(\alpha\)

\(\rho_{11}\) \(\beta\) \(\delta\) \(\alpha\) \(\gamma\)

\(\rho_{12}\) \(\beta\) \(\delta\) \(\gamma\) \(\alpha\)

\(\rho_{13}\) \(\gamma\) \(\alpha\) \(\beta\) \(\delta\)

\(\rho_{14}\) \(\gamma\) \(\alpha\) \(\delta\) \(\beta\)

\(\rho_{15}\) \(\gamma\) \(\beta\) \(\alpha\) \(\delta\)

\(\rho_{16}\) \(\gamma\) \(\beta\) \(\delta\) \(\alpha\)

[Table 3-2] Transformation table for \(\rho_i\) (2)
	\(\rho_i(\alpha)\)	\(\rho_i(\beta)\)	\(\rho_i(\gamma)\)	\(\rho_i(\delta)\)
\(\rho_{9}\)	\(\beta\)	\(\gamma\)	\(\alpha\)	\(\delta\)
\(\rho_{10}\)	\(\beta\)	\(\gamma\)	\(\delta\)	\(\alpha\)
\(\rho_{11}\)	\(\beta\)	\(\delta\)	\(\alpha\)	\(\gamma\)
\(\rho_{12}\)	\(\beta\)	\(\delta\)	\(\gamma\)	\(\alpha\)
\(\rho_{13}\)	\(\gamma\)	\(\alpha\)	\(\beta\)	\(\delta\)
\(\rho_{14}\)	\(\gamma\)	\(\alpha\)	\(\delta\)	\(\beta\)
\(\rho_{15}\)	\(\gamma\)	\(\beta\)	\(\alpha\)	\(\delta\)
\(\rho_{16}\)	\(\gamma\)	\(\beta\)	\(\delta\)	\(\alpha\)

[Table 3-2] Transformation table for \(\rho_i\) (3)
\(\rho_i(\alpha)\) \(\rho_i(\beta)\) \(\rho_i(\gamma)\) \(\rho_i(\delta)\)

\(\rho_{17}\) \(\gamma\) \(\delta\) \(\alpha\) \(\beta\)

\(\rho_{18}\) \(\gamma\) \(\delta\) \(\beta\) \(\alpha\)

\(\rho_{19}\) \(\delta\) \(\alpha\) \(\beta\) \(\gamma\)

\(\rho_{20}\) \(\delta\) \(\alpha\) \(\gamma\) \(\beta\)

\(\rho_{21}\) \(\delta\) \(\beta\) \(\alpha\) \(\gamma\)

\(\rho_{22}\) \(\delta\) \(\beta\) \(\gamma\) \(\alpha\)

\(\rho_{23}\) \(\delta\) \(\gamma\) \(\alpha\) \(\beta\)

\(\rho_{24}\) \(\delta\) \(\gamma\) \(\beta\) \(\alpha\)

[Table 3-2] Transformation table for \(\rho_i\) (3)
	\(\rho_i(\alpha)\)	\(\rho_i(\beta)\)	\(\rho_i(\gamma)\)	\(\rho_i(\delta)\)
\(\rho_{17}\)	\(\gamma\)	\(\delta\)	\(\alpha\)	\(\beta\)
\(\rho_{18}\)	\(\gamma\)	\(\delta\)	\(\beta\)	\(\alpha\)
\(\rho_{19}\)	\(\delta\)	\(\alpha\)	\(\beta\)	\(\gamma\)
\(\rho_{20}\)	\(\delta\)	\(\alpha\)	\(\gamma\)	\(\beta\)
\(\rho_{21}\)	\(\delta\)	\(\beta\)	\(\alpha\)	\(\gamma\)
\(\rho_{22}\)	\(\delta\)	\(\beta\)	\(\gamma\)	\(\alpha\)
\(\rho_{23}\)	\(\delta\)	\(\gamma\)	\(\alpha\)	\(\beta\)
\(\rho_{24}\)	\(\delta\)	\(\gamma\)	\(\beta\)	\(\alpha\)

Strictly speaking, each \(\rho_i\) was defined to send \(v\) to a conjugate root, but through \(v\) the action propagates to the original roots \(\{\alpha,\beta,\gamma,\delta\}\) as well—this is a pleasant feature of the construction.

In short, the “mapping-like” behavior of \(\rho_i\) makes the name automorphism feel appropriate. The flavor differs from the permutation operators \(\sigma_i\)—even though, in fact, they are the same. We summarize below.

\( \quad\{\rho_1,\rho_2,\ldots,\rho_{24}\} = Gal(F_0(v)/F_0) \cong S_4\)

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⇨

\( i \backslash j \)	\(1 \)	\(2 \)	\(3 \)	\(4 \)	\(5 \)	\(6 \)	\(7 \)	\(8 \)	\(9 \)	\({10} \)	\({11} \)	\({12} \)	\({13} \)	\({14} \)	\({15} \)	\({16} \)	\({17} \)	\({18} \)	\({19} \)	\({20} \)	\({21} \)	\({22} \)	\({23} \)	\({24}\)
\(1\)	\(1 \)	\(2 \)	\(3 \)	\(4 \)	\(5 \)	\(6 \)	\(7 \)	\(8 \)	\(9 \)	\({10} \)	\({11} \)	\({12} \)	\({13} \)	\({14} \)	\({15} \)	\({16} \)	\({17} \)	\({18} \)	\({19} \)	\({20} \)	\({21} \)	\({22} \)	\({23} \)	\({24}\)
\(2\)	\(2 \)	\(1 \)	\(5 \)	\(6 \)	\(3 \)	\(4 \)	\(8 \)	\(7 \)	\({11} \)	\({12} \)	\(9 \)	\({10} \)	\({19} \)	\({20} \)	\({21} \)	\({22} \)	\({23} \)	\({24}\)	\({13} \)	\({14} \)	\({15} \)	\({16} \)	\({17} \)	\({18}\)
\(3 \)	\(3 \)	\(4 \)	\(1 \)	\(2 \)	\(6 \)	\(5 \)	\({13} \)	\({14} \)	\({15} \)	\({16} \)	\({17} \)	\({18} \)	\(7 \)	\(8 \)	\(9 \)	\({10} \)	\({11} \)	\({12} \)	\({20} \)	\({19} \)	\({23} \)	\({24}\)	\({21} \)	\({22} \)
\(4\)	\(4 \)	\(3 \)	\(6 \)	\(5 \)	\(1 \)	\(2 \)	\({14} \)	\({13} \)	\({17} \)	\({18} \)	\({15} \)	\({16} \)	\({20} \)	\({19} \)	\({23} \)	\({24}\)	\({21} \)	\({22} \)	\(7 \)	\(8 \)	\(9 \)	\({10} \)	\({11} \)	\({12} \)
\(5 \)	\(5 \)	\(6 \)	\(2 \)	\(1 \)	\(4 \)	\(3 \)	\({19} \)	\({20} \)	\({21} \)	\({22} \)	\({23} \)	\({24}\)	\(8 \)	\(7 \)	\({11} \)	\({12} \)	\(9 \)	\({10} \)	\({14} \)	\({13} \)	\({17} \)	\({18} \)	\({15} \)	\({16} \)
\(6 \)	\(6 \)	\(5 \)	\(4 \)	\(3 \)	\(2 \)	\(1 \)	\({20} \)	\({19} \)	\({23} \)	\({24}\)	\({21} \)	\({22} \)	\({14} \)	\({13} \)	\({17} \)	\({18} \)	\({15} \)	\({16} \)	\(8 \)	\(7 \)	\({11} \)	\({12} \)	\(9 \)	\({10} \)
\(7 \)	\(7 \)	\(8 \)	\(9 \)	\({10} \)	\({11} \)	\({12} \)	\(1 \)	\(2 \)	\(3 \)	\(4 \)	\(5 \)	\(6 \)	\({15} \)	\({16} \)	\({13} \)	\({14} \)	\({18} \)	\({17} \)	\({21} \)	\({22} \)	\({19} \)	\({20} \)	\({24}\)	\({23} \)
\(8 \)	\(8 \)	\(7 \)	\({11} \)	\({12} \)	\(9 \)	\({10} \)	\(2 \)	\(1 \)	\(5 \)	\(6 \)	\(3 \)	\(4 \)	\({21} \)	\({22} \)	\({19} \)	\({20} \)	\({24}\)	\({23} \)	\({15} \)	\({16} \)	\({13} \)	\({14} \)	\({18} \)	\({17} \)
\(9 \)	\(9 \)	\({10} \)	\(7 \)	\(8 \)	\({12} \)	\({11} \)	\({15} \)	\({16} \)	\({13} \)	\({14} \)	\({18} \)	\({17} \)	\(1 \)	\(2 \)	\(3 \)	\(4 \)	\(5 \)	\(6 \)	\({22} \)	\({21} \)	\({24}\)	\({23} \)	\({19} \)	\({20} \)
\({10} \)	\({10} \)	\(9 \)	\({12} \)	\({11} \)	\(7 \)	\(8 \)	\({16} \)	\({15} \)	\({18} \)	\({17} \)	\({13} \)	\({14} \)	\({22} \)	\({21} \)	\({24}\)	\({23} \)	\({19} \)	\({20} \)	\(1 \)	\(2 \)	\(3 \)	\(4 \)	\(5 \)	\(6 \)
\({11} \)	\({11} \)	\({12} \)	\(8 \)	\(7 \)	\({10} \)	\(9 \)	\({21} \)	\({22} \)	\({19} \)	\({20} \)	\({24}\)	\({23} \)	\(2 \)	\(1 \)	\(5 \)	\(6 \)	\(3 \)	\(4 \)	\({16} \)	\({15} \)	\({18} \)	\({17} \)	\({13} \)	\({14} \)
\({12} \)	\({12} \)	\({11} \)	\({10} \)	\(9 \)	\(8 \)	\(7 \)	\({22} \)	\({21} \)	\({24}\)	\({23} \)	\({19} \)	\({20} \)	\({16} \)	\({15} \)	\({18} \)	\({17} \)	\({13} \)	\({14} \)	\(2 \)	\(1 \)	\(5 \)	\(6 \)	\(3 \)	\(4 \)
\({13} \)	\({13} \)	\({14} \)	\({15} \)	\({16} \)	\({17} \)	\({18} \)	\(3 \)	\(4 \)	\(1 \)	\(2 \)	\(6 \)	\(5 \)	\(9 \)	\({10} \)	\(7 \)	\(8 \)	\({12} \)	\({11} \)	\({23} \)	\({24}\)	\({20} \)	\({19} \)	\({22} \)	\({21} \)
\({14} \)	\({14} \)	\({13} \)	\({17} \)	\({18} \)	\({15} \)	\({16} \)	\(4 \)	\(3 \)	\(6 \)	\(5 \)	\(1 \)	\(2 \)	\({23} \)	\({24}\)	\({20} \)	\({19} \)	\({22} \)	\({21} \)	\(9 \)	\({10} \)	\(7 \)	\(8 \)	\({12} \)	\({11} \)
\({15} \)	\({15} \)	\({16} \)	\({13} \)	\({14} \)	\({18} \)	\({17} \)	\(9 \)	\({10} \)	\(7 \)	\(8 \)	\({12} \)	\({11} \)	\(3 \)	\(4 \)	\(1 \)	\(2 \)	\(6 \)	\(5 \)	\({24}\)	\({23} \)	\({22} \)	\({21} \)	\({20} \)	\({19} \)
\({16} \)	\({16} \)	\({15} \)	\({18} \)	\({17} \)	\({13} \)	\({14} \)	\({10} \)	\(9 \)	\({12} \)	\({11} \)	\(7 \)	\(8 \)	\({24}\)	\({23} \)	\({22} \)	\({21} \)	\({20} \)	\({19} \)	\(3 \)	\(4 \)	\(1 \)	\(2 \)	\(6 \)	\(5 \)
\({17} \)	\({17} \)	\({18} \)	\({14} \)	\({13} \)	\({16} \)	\({15} \)	\({23} \)	\({24}\)	\({20} \)	\({19} \)	\({22} \)	\({21} \)	\(4 \)	\(3 \)	\(6 \)	\(5 \)	\(1 \)	\(2 \)	\({10} \)	\(9 \)	\({12} \)	\({11} \)	\(7 \)	\(8 \)
\({18} \)	\({18} \)	\({17} \)	\({16} \)	\({15} \)	\({14} \)	\({13} \)	\({24}\)	\({23} \)	\({22} \)	\({21} \)	\({20} \)	\({19} \)	\({10} \)	\(9 \)	\({12} \)	\({11} \)	\(7 \)	\(8 \)	\(4 \)	\(3 \)	\(6 \)	\(5 \)	\(1 \)	\(2 \)
\({19} \)	\({19} \)	\({20} \)	\({21} \)	\({22} \)	\({23} \)	\({24}\)	\(5 \)	\(6 \)	\(2 \)	\(1 \)	\(4 \)	\(3 \)	\({11} \)	\({12} \)	\(8 \)	\(7 \)	\({10} \)	\(9 \)	\({17} \)	\({18} \)	\({14} \)	\({13} \)	\({16} \)	\({15} \)
\({20} \)	\({20} \)	\({19} \)	\({23} \)	\({24}\)	\({21} \)	\({22} \)	\(6 \)	\(5 \)	\(4 \)	\(3 \)	\(2 \)	\(1 \)	\({17} \)	\({18} \)	\({14} \)	\({13} \)	\({16} \)	\({15} \)	\({11} \)	\({12} \)	\(8 \)	\(7 \)	\({10} \)	\(9 \)
\({21} \)	\({21} \)	\({22} \)	\({19} \)	\({20} \)	\({24}\)	\({23} \)	\({11} \)	\({12} \)	\(8 \)	\(7 \)	\({10} \)	\(9 \)	\(5 \)	\(6 \)	\(2 \)	\(1 \)	\(4 \)	\(3 \)	\({18} \)	\({17} \)	\({16} \)	\({15} \)	\({14} \)	\({13} \)
\({22} \)	\({22} \)	\({21} \)	\({24}\)	\({23} \)	\({19} \)	\({20} \)	\({12} \)	\({11} \)	\({10} \)	\(9 \)	\(8 \)	\(7 \)	\({18} \)	\({17} \)	\({16} \)	\({15} \)	\({14} \)	\({13} \)	\(5 \)	\(6 \)	\(2 \)	\(1 \)	\(4 \)	\(3 \)
\({23} \)	\({23} \)	\({24}\)	\({20} \)	\({19} \)	\({22} \)	\({21} \)	\({17} \)	\({18} \)	\({14} \)	\({13} \)	\({16} \)	\({15} \)	\(6 \)	\(5 \)	\(4 \)	\(3 \)	\(2 \)	\(1 \)	\({12} \)	\({11} \)	\({10} \)	\(9 \)	\(8 \)	\(7 \)
\({24}\)	\({24}\)	\({23} \)	\({22} \)	\({21} \)	\({20} \)	\({19} \)	\({18} \)	\({17} \)	\({16} \)	\({15} \)	\({14} \)	\({13} \)	\({12} \)	\({11} \)	\({10} \)	\(9 \)	\(8 \)	\(7 \)	\(6 \)	\(5 \)	\(4 \)	\(3 \)	\(2 \)	\(1 \)