Techniques of Solving Equations à la Galois
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1st upload: 2023/06/17
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[3-4] The minimal polynomial \(g_0(x)\), the four roots \(\{\alpha,\beta,\gamma,\delta\}\), and the polynomials in \(v_i\)
Since \(V(x)\) is irreducible over \(F_0\), we may take \(V(x)\) itself as the minimal polynomial \(g_0(x)\) of \(v\).Moreover, because \(V(x)\) has \(v=v_1\) as a root, it is of course the case that \(g_0(v)=0\).
\begin{align} g_0(x)&:minimal \ polynomial \ of \ v \ on \ F_0 \notag \\ \notag \\ V(x) \equiv g_0(x)&= x^{24}-160x^{20}+5440x^{18}+.....+4691625312256 \\ \notag \\ g_0(v)&=v^{24}-160v^{20}+5440v^{18}+.....+4691625312256=0 \\ \end{align}
Polynomial expressions of \(\{\alpha,\beta,\gamma,\delta\}\) in terms of \(v\)
To determine the four roots \(\{\alpha,\beta,\gamma,\delta\}\) of \(f(x)\), we first need the reciprocal \(V'(x)^{-1}\).By the same method as in Chapters 1 and 2, we obtain (4.4), whose coefficients are quite complicated. Note that reduction modulo \(g_0(v)\) is essential during the computation.
\begin{align} V^{'}(v)&=24v^{23}-3200v^{19}+97920v^{17}+481280v^{15}+10357760v^{13}\notag \\ &+304809984v^{11}-295936000v^9+12163317760v^7+213195325440v^5\notag \\ &+1644537184256v^3+1400183193600v \\ \end{align}
\begin{align} V^{'}(v)^{-1}&=\frac{36650658007597998057527901863523813134582549442939859625}{1298927928510468889470594879182809105052836606048849589825447492902816344937857024}v^{23} \notag \\ \notag \\ &-\frac{1320201362648416679393021145878628722365418769923}{13608270331256762059701528955837861614568450918781199465890427037556932608}v^{21} \notag \\ \notag \\ &+.......... \notag \\ \notag \\ &+\frac{(226184448292414813563319989189735356244047283289116150200621045}{29730100902796986911100651836764734765308455033466711192904224233310310629376}v^3 \notag \\ \notag \\ &-\frac{1359699243330685835420943609764282003859074131280909865754805267}{39640134537062649214800869115686313020411273377955614923872298977747080839168}v \\ \end{align}
Substituting this reciprocal \(V'(x)^{-1}\) together with \(P_{\alpha}(v)\) obtained in the previous section into (4.5), we get the polynomial expression of \(\alpha\) in terms of \(v\).
\begin{align} \alpha&=\left.\frac{P_\alpha(x)}{V'(x)}\right|_{x=v}=P_{\alpha}(v) \cdot V^{'}(v)^{-1} & \beta&=\left.\frac{P_\beta(x)}{V'(x)}\right|_{x=v}=P_{\beta}(v) \cdot V^{'}(v)^{-1} \\ \gamma&=\left.\frac{P_\gamma(x)}{V'(x)}\right|_{x=v}=P_{\gamma}(v) \cdot V^{'}(v)^{-1} & \delta&=\left.\frac{P_\delta(x)}{V'(x)}\right|_{x=v}=P_{\delta}(v) \cdot V^{'}(v)^{-1} \notag \\ \end{align}
Carrying out the same computation for the other roots, we obtain the following very complicated polynomial expressions for \(\{\alpha,\beta,\gamma,\delta\}\).
\begin{align} \alpha&= \frac{801167701943012874015343807}{10126546386824616812436636833146824818688}v^{23}+\frac{51207699710669004924125}{199474971178044691573821786887815168}v^{22} \notag \\ &\quad .......... \notag \\ &-\frac{1279063375083309131586881879157101}{11886064249859873624873982160300416}v-\frac{2718803338720300088760700554765}{3043746508454051079922817793088} \\ \notag \\ \beta&=-\frac{801167701943012874015343807}{3375515462274872270812212277715608272896}v^{23}-\frac{51207699710669004924125}{199474971178044691573821786887815168}v^{22} \notag \\ &\quad .......... \notag \\ &-\frac{2682958041536648743371112174276371}{3962021416619957874957994053433472}v+\frac{2718803338720300088760700554765}{3043746508454051079922817793088} \\ \notag \\ \gamma&=\frac{801167701943012874015343807}{3375515462274872270812212277715608272896}v^{23}-\frac{51207699710669004924125}{199474971178044691573821786887815168}v^{22} \notag \\ &\quad .......... \notag \\ &+\frac{2682958041536648743371112174276371}{3962021416619957874957994053433472}v+\frac{2718803338720300088760700554765}{3043746508454051079922817793088} \\ \notag \\ \delta&=-\frac{801167701943012874015343807}{10126546386824616812436636833146824818688}v^{23}+\frac{51207699710669004924125}{199474971178044691573821786887815168}v^{22} \notag \\ &\quad .......... \notag \\ &+\frac{1279063375083309131586881879157101}{11886064249859873624873982160300416}v-\frac{2718803338720300088760700554765}{3043746508454051079922817793088} \\ \end{align}
Polynomial expressions of \(\{v_i \mid i=1,2,\ldots,23,24\}\) in terms of \(v\)
Substituting the expressions of \(\{\alpha,\beta,\gamma,\delta\}\) in terms of \(v\) into (2.3) for \(v_i\), we obtain polynomial expressions for each \(v_i\) in terms of \(v\).\begin{align} v_1&=v \\ \notag \\ 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 \\ v_{23}&=-\frac{801167701943012874015343807}{2531636596706154203109159208286706204672}v^{23}-\frac{51207699710669004924125}{99737485589022345786910893443907584}v^{22} \notag \\ &\quad .......... \notag \\ &-\frac{4663968749846627680850109200993107}{2971516062464968406218495540075104}v+\frac{2718803338720300088760700554765}{1521873254227025539961408896544} \\ \notag \\ v_{24}&=-v \\ \end{align}
Substituting the computed values \(\{v_1,v_2,\ldots,v_{24}\}\) into \(g_0(x)\), we find in each case that the result is zero, so they are indeed roots. Hence the \(v_i\) are mutually conjugate roots of \(g_0(x)\).
\begin{align} &g_0(v_1)=g_0(v_2)=....=g_0(v_{23})=g_0(v_{24})=0\quad ( \ mod \ g_0(v) \ )\\ \end{align}
From the above we conclude that all \(v_i\ [i=1,2,\ldots,24]\) are conjugate roots of \(g_0(x)\).
Let us revisit the conditions [1] and [2] for a Galois extension stated in Section 2.
Condition [2]: Since \(\{v_1,v_2,...,v_6\}\) are expressed as polynomials in \(v\), the field \(F_0(v)\) contains all the roots of the minimal polynomial \(g_0(x)\). Therefore \(F_0(v)/F_0\) is a normal extension.
Consequently, \(F_0(v)\) satisfies conditions [1] and [2], so \(F_0(v)/F_0\) is a Galois extension.
Let \( Gal(F_0(v)/F_0) \) denote the Galois group of the extension \(F_0(v)/F_0\). Then we have:
\begin{align} \# \ Gal(F_0(v)/F_0) =[ \ F_0(v):F_0 \ ] =24=deg \ \bigl( \ g_0(x) \ \bigr) \notag \\ \end{align}