... question
<#1720#>Other equally natural questions are
  • to solve f by radicals, in case it is possible, and
  • the <#3#>Galois inverse problem<#3#> of determining, given a finite group G, a polynomial f∈k[T] for which G is its Galois group.
Such questions are just mentioned here but not discussed in this book. <#1720#>
...Let
<#1721#>For this theory, compare Burnside W., <#14#>Theory of groups of finite order<#14#>, Cambridge Universiy Press (1911) Ch. XII<#1721#>
... that
<#1722#>In particular: #math87##tex2html_wrap_inline4942#C1 = {{#tex2html_wrap_inline4943#}} and #math88##tex2html_wrap_inline4947#Cs = {G}.<#1722#>
... elements
<#245#>So that, in particular #math119#m = α1. <#245#>
... notation
<#1723#>In particular #math122#H∈#tex2html_wrap_inline5103#C, #math123#H'∈#tex2html_wrap_inline5106#C′ and #math124#H⊂H'.<#1723#>
...ρj
<#1726#> <#271#>Id est<#271#> the number of elements #math132#H'∈#tex2html_wrap_inline5140#Cj which satisfy #math133#ρj(h)(H') = H'#tex2html_wrap_inline5142#h∈H.<#1726#>
... the
<#298#>Burnside W., op. cit., p. 238<#298#>
... elements
<#1728#>So that, in particular #math152#mij = a(1)ji. <#1728#>
... roots
<#1790#>More precisely, let #math165#t∈#tex2html_wrap_inline5304# and defined #math166#βi : = αt(i) for each i; then for each #math167#σ∈G(Kf/k) we have

#math168#

σ(βi) = σ(αt(i)) = αsσt(i) = βt-1sσt(i)#tex2html_wrap_indisplay5309#i;

so a different enumeration of roots simply give an equivalent representation of G(Kf/k).<#1790#>
...]
<#1791#>In fact if #math218#x∈#tex2html_wrap_inline5487#(H), then there are #math219#α, β∈#tex2html_wrap_inline5489#A, #math220#β≠ 0 such that #math221#x = #tex2html_wrap_inline5492#.

It is sufficient to define

#math222#

b : = #tex2html_wrap_indisplay5494#σ(β)∈#tex2html_wrap_indisplay5495#S #tex2html_wrap_indisplay5496# {0},;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;a : = α#tex2html_wrap_indisplay5497#σ(β)∈#tex2html_wrap_indisplay5500#AH

in order to have #math223#x = #tex2html_wrap_inline5502#. <#1791#>
...
<#1746#>A. Cauchy <#744#>Usage des fonctions interpolaires dans ls determination des fonctions symmetriques des racines d'une équation algébrique donnée<#744#> C.R. Acad. Sci. Paris <#745#>11<#745#> (1840) p.933,

In: A. Cauchy <#746#>Oeuvres<#746#> t. V, Gauthier--Villars (1882) Paris , pp. 476--7.<#1746#>

...0]$.
<#752#>A. Cauchy, op. cit., pp. 474--5.<#752#>
...
<#1753#>Valibouze A., <#1016#>Théorie de Galois constructive<#1016#>, Mémoir d'Habilitation, Paris 6 (1998) p. 23.<#1753#>
...which
<#1761#>$<#1225#>A<#1225#>$ is noetherian.<#1761#>
...n≤11
<#1777#>Here I report their result for n = 4; for #math366#5≤n≤11 I refer to the survey Valibouze A., <#1633#>Computation of the Galois Groups of the Resolvent Factors for the Deirect and Inverse Galois problems<#1633#> L. N. Comp. Sci. <#1634#>948<#1634#> (1995), 456--468, Springer, and to the LITP reports quoted there.<#1777#>
... are
<#1671#>where #math370#u1, u2, u3 are distinct and non zero values.<#1671#>
...;SPMnbsp;
<#1789#>one could similarly use #math398##tex2html_wrap_inline6267#LΘi, f, i∈{1, 2, 3, 7}.<#1789#>