Techniques of Solving Equations à la Galois
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[3-3] Introducing the polynomials \(V(x), \ P_{\alpha}(x), \ P_{\beta}(x), \ P_{\gamma}(x), \ P_{\delta}(x)\)
As before, we introduce five auxiliary polynomials. \(V(x)\) will turn out to be the “minimal polynomial.”The polynomials \(\{P_{\alpha}(x), P_{\beta}(x), P_{\gamma}(x), P_{\delta}(x)\}\) are used to express the roots \(\{ \ \alpha,\beta,\gamma,\delta \ \}\) in terms of \(v\).
This time, exactly as in Chapter 2, we convert these five polynomials into polynomials over the base field \(F_0\).
\begin{align} V(x)&\equiv \displaystyle \prod_{i=1}^{24}\sigma_i(x-v) \\ P_\alpha(x)&\equiv V(x)\cdot \Bigl[ \ \sum_{i=1}^{24} \sigma_i(\frac{\alpha }{x-v}) \ \Bigr] , \ P_{\beta}(x)\equiv V(x)\cdot \Bigl[ \ \sum_{i=1}^{24} \sigma_i(\frac{\beta }{x-v}) \ \Bigr] \\ P_{\gamma}(x)&\equiv V(x)\cdot \Bigl[ \ \sum_{i=1}^{24} \sigma_i(\frac{\gamma }{x-v}) \ \Bigr] , \ P_{\delta}(x)\equiv V(x)\cdot \Bigl[ \ \sum_{i=1}^{24} \sigma_i(\frac{\delta }{x-v}) \ \Bigr] \\ \end{align}
We illustrate the method with \(V(x)\). Substitute (2.3) into the \(v_i\) in (3.1) and expand.
\begin{align} &V(x)=(x-v_{1})(x-v_{2})(x-v_{3}).....(x-v_{22})(x-v_{23})(x-v_{24}) \\ &\qquad \Downarrow \notag \\ &V(x,\alpha,\beta,\gamma,\delta)= x^{24}-60(\delta +\gamma +\beta +\alpha )x^{23} +......\notag \\ \end{align}
Nevertheless—following the same approach as in Chapter 2—if we reduce \(V(x,\alpha,\beta,\gamma,\delta)\) successively by the relations (1.7), namely \(\{ r_4(\delta),\ r_3(\gamma),\ r_2(\beta), \ r_1(\alpha) \}\) in this order, we obtain a polynomial over \(F_0\).
\begin{align} V(x)&=x^{24}-160x^{20}+5440x^{18}+30080x^{16}+739840x^{14} \notag \\ &+25400832x^{12}-29593600x^{10}+1520414720x^8 \notag \\ &+35532554240x^6+411134296064x^4+700091596800x^2 \notag \\ &+4691625312256 \qquad \in \ F_0[x]\\ \end{align}
It is striking that \(V(x,\alpha,\beta,\gamma,\delta)\) collapses all at once to a polynomial over \(F_0\).
Incidentally, the \(V(x)\) in (3.6) is called the Galois resolvent.
We compute \(\{ P_\alpha(x),P_\beta(x),P_\gamma(x),P_\delta(x)\}\) in the same manner as \(V(x)\); see (3.2)–(3.3).
As an example, here is the transformation for \( P_\alpha(x)\).
\begin{align} &P_\alpha(x)\equiv V(x)\cdot \Bigl[ \ \sum_{i=1}^{24} \sigma_i(\frac{\alpha }{x-v}) \ \Bigr] \notag \\ &\qquad \Downarrow \notag \\ &P_\alpha(x,\alpha,\beta,\gamma,\delta)=c_{23}x^{23}+c_{22}x^{22}+c_{21}x^{21}+....+c_{3}x^{3}+c_{2}x^{2}+c_{1}x+c_0 \notag \\ \notag \\ &c_{23}=6(\alpha+\beta+\gamma+\delta) \notag \\ &c_{22}=-6 \left( 59\delta^2+114\gamma\delta+114\beta\delta+114\alpha\delta+59\gamma^2 \right. \notag \\ &\qquad \left. +114\beta\gamma+114\alpha\gamma+59\beta^2+114\alpha\beta+59\alpha^2 \right) \notag \\ &c_{21}=2 \left(4953\delta^3+14297\gamma\delta^2+14297\beta\delta^2+14297\alpha\delta^2+14297\gamma^2\delta \right. \notag \\ &\qquad +28056\beta\gamma\delta+28056\alpha\gamma\delta+14297\beta^2\delta+28056\alpha\beta\delta+14297\alpha^2\delta \notag \\ &\qquad +4953\gamma^3+14297\beta\gamma^2+14297\alpha\gamma^2+14297\beta^2\gamma+28056\alpha\beta\gamma \notag \\ &\qquad \left. +14297\alpha^2\gamma+4953\beta^3+14297\alpha\beta^2+14297\alpha^2\beta+4953\alpha^3 \right) \notag \\ & \qquad \qquad ......... \notag \\ \notag \\ &\qquad \Downarrow \quad \bbox[#FFFF00]{ mod(r_4(\delta)),\ mod(r_3(\gamma)),\ mod(r_2(\beta)), \ mod(r_1(\alpha)) } \notag \\ \notag \\ &c_{23}=0 \quad c_{22}=0 \quad c_{21}=-192 \quad ..... \notag \\ \end{align}
In the transformations above, it may look as though we reduce each coefficient \(c_i\) by (1.7) using \(\{ r_4(\delta),\ r_3(\gamma),\ r_2(\beta), \ r_1(\alpha) \}\). In practice, however, we reduce the entire polynomial \(P_\alpha(x,\alpha,\beta,\gamma,\delta)\) at once by (1.7) to obtain \( P_\alpha(x)\).
The resulting polynomials \(\{ P_\alpha(x),P_\beta(x),P_\gamma(x),P_\delta(x)\}\) are as follows.
\begin{align} P_\alpha(x)&=-192x^{21}+13824x^{18}-1024x^{17}+30720x^{16}-983040x^{15} \notag \\ &+2433024x^{14}-2768896x^{13}+105713664x^{12}-120504320x^{11} \notag \\ &-35880960x^{10}-1756086272x^9+9626910720x^8-19995033600x^7 \notag \\ &+176799350784x^6 -301266108416x^5+2423048110080x^4 \notag \\ &-1707917967360x^3+3237850644480x^2 \notag \\ &-2155349016576x+31486733451264 \\ \notag \\ P_\beta(x)&=192x^{21}-640x^{20}-8832x^{18}+1024x^{17}+148480x^{16}+983040x^{15} \notag \\ &+99328x^{14}+2768896x^{13}-12331008x^{12}+120504320x^{11} \notag \\ &-306667520x^{10}+1756086272x^9-4554096640x^8+19995033600x^7 \notag \\ &+109187923968x^6+301266108416x^5+953541591040x^4 \notag \\ &+1707917967360x^3+5688463196160x^2 \notag \\ &+2155349016576x+18138807140352 \\ \notag \\ P_\gamma(x)&=192x^{21}+640x^{20}+8832x^{18}+1024x^{17}-148480x^{16}+983040x^{15}\notag \\ &-99328x^{14}+2768896x^{13}+12331008x^{12}+120504320x^{11} \notag \\ &+306667520x^{10}+1756086272x^{9}+4554096640x^{8}+19995033600x^{7} \notag \\ &-109187923968x^{6}+301266108416x^{5}-953541591040x^{4} \notag \\ &+1707917967360x^{3}-5688463196160x^{2} \notag \\ &+2155349016576x-18138807140352 \\ \notag \\ P_\delta(x)&=-192x^{21}-13824x^{18}-1024x^{17}-30720x^{16}-983040x^{15} \notag \\ &-2433024x^{14}-2768896x^{13}-105713664x^{12}-120504320x^{11} \notag \\ &+35880960x^{10}-1756086272x^9-9626910720x^8-19995033600x^7 \notag \\ &-176799350784x^6-301266108416x^5-2423048110080x^4 \notag \\ &-1707917967360x^3-3237850644480x^2 \notag \\ &-2155349016576x-31486733451264 \\ \end{align}