Editing Solvable group (section)

== Motivation ==
Historically, the word "solvable" arose from [[Galois theory]] and the proof of the general unsolvability of [[quintic]] equations. Specifically, a [[polynomial equation]] is solvable in [[Nth root|radicals]] if and only if the corresponding [[Galois group]] is solvable<ref>{{Cite book|last=Milne|url=https://www.jmilne.org/math/CourseNotes/FT.pdf|title=Field Theory|pages=45}}</ref> (note this theorem holds only in [[characteristic of a field|characteristic]] 0). This means associated to a polynomial <math>f \in F[x]</math> there is a tower of field extensions<blockquote><math>F = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=K</math></blockquote>such that

# <math>F_i = F_{i-1}[\alpha_i]</math> where <math>\alpha_i^{m_i} \in F_{i-1}</math>, so <math>\alpha_i</math> is a solution to the equation <math>x^{m_i} - a</math> where <math>a \in F_{i-1}</math>
# <math>F_m</math> contains a [[splitting field]] for <math>f(x)</math>

=== Example ===
The smallest Galois field extension of <math>\mathbb{Q}</math> containing the element<blockquote><math>a = \sqrt[5]{\sqrt{2} + \sqrt{3}}</math></blockquote>gives a solvable group. The associated field extensions<blockquote><math>\mathbb{Q} 
\subseteq \mathbb{Q}(\sqrt{2})  
\subseteq \mathbb{Q}(\sqrt{2}, \sqrt{3})  
\subseteq \mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right)
\subseteq \mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}, a\right)</math></blockquote>give a solvable group of Galois extensions containing the following [[composition factor]]s (where <math>1</math> is the identity permutation).

* <math>\mathrm{Aut}\left(\mathbb{Q(\sqrt{2})}\right/\mathbb{Q}) \cong \mathbb{Z}/2 </math> with group action <math>f\left(\pm\sqrt{2}\right) = \mp\sqrt{2}, \ f^2 = 1</math>, and [[Minimal polynomial (field theory)|minimal polynomial]] <math>x^2 - 2</math>
* <math>\mathrm{Aut}\left(\mathbb{Q(\sqrt{2},\sqrt{3})}\right/\mathbb{Q(\sqrt{2})}) \cong \mathbb{Z}/2 </math> with group action <math>g\left(\pm\sqrt{3}\right) = \mp\sqrt{3} ,\ g^2 = 1</math>, and minimal polynomial <math>x^2 - 3</math>
* <math>\mathrm{Aut}\left(
\mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right)/
\mathbb{Q}(\sqrt{2}, \sqrt{3})
\right)
\cong \mathbb{Z}/4 </math> with group action <math>h^n\left(e^{2im\pi/5}\right) = e^{2(n+1)mi\pi/5} , \  0 \leq n \leq 3, \ h^4 = 1</math>, and minimal polynomial <math>x^4 + x^3+x^2+x+1 = (x^5 - 1)/(x-1)</math> containing the 5th roots of unity excluding <math>1</math>
* <math>\mathrm{Aut}\left(
\mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}, a\right)/
\mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right)
\right)
\cong \mathbb{Z}/5 </math> with group action <math>j^l(a) = e^{2li\pi/5}a, \ j^5 = 1</math>, and minimal polynomial <math>x^5 - \left(\sqrt{2} + \sqrt{3}\right)</math>

Each of the defining group actions (for example, <math>fgh^3j^4 </math>) changes a single extension while keeping all of the other extensions fixed. The 80 group actions are the set <math>\{f^ag^bh^nj^l,\ 0 \leq a, b \leq 1,\ 0 \leq n \leq 3,\ 0 \leq l \leq 4 \}</math>.

This group is not [[Abelian group|abelian]].  For example, <math>hj(a) = h(e^{2i\pi/5}a) = e^{4i\pi/5}a </math>, whilst <math>jh(a) = j(a) = e^{2i\pi/5}a</math>, and in fact, <math>jh = hj^3</math>. 

It is isomorphic to <math>(\mathbb{Z}_5 \rtimes_\varphi \mathbb{Z}_4) \times (\mathbb{Z}_2 \times \mathbb{Z}_2) </math>, where <math>\varphi_h(j) = hjh^{-1} = j^2 </math>, defined using the [[semidirect product]] and [[Direct product of groups|direct product]] of the [[cyclic group]]s.  <math>\mathbb{Z}_4 </math> is not a normal subgroup.