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\parbox[b]{3.8in}{
  \footnotesize\url{http://drorbn.net/cumc20}
  \newline\bf
  \href{\myurl/}{Dror Bar-Natan}:
  \href{\myurl/Talks/}{Talks}:
  \href{\myurl/Talks/Sydney-1708/}{CUMC-200820}:
}
\hfill\parbox[b]{4.0in}{
  \null\hfill{\Huge\bf Nobody Solves the Quintic}
}

\begin{multicols}{2}

{\bf Abstract.} Everybody knows that nobody can solve the quintic. Indeed this insolubility is a well known hard theorem, the high point of a full-semester course on Galois theory, often taken in one's 3rd or 4th year of university mathematics. I'm not sure why so few know that the same theorem can be proven in about 15 minutes using *very* basic and easily understandable topology, accessible to practically anyone.

\rule{\columnwidth}{0.5pt}

\textbf{Definition.} The commutator of two elements $x$ and $y$ in a group $G$ is $[x,y] \coloneqq xyx^{-1} y^{-1}$.

\textbf{Example 0.} In $\mathbb Z$, $[m,n]=0$.

\textbf{Example 1.} In $S_3$, $[(12), (23)] = (12) (23) (12)^{-1} (23)^{-1}=(123)$ and in general in $S_{\geq 3}$,
\[ [(ij),(jk)]=(ijk). \]

\textbf{Example 2.} In $S_{\geq 4}$,
\[ [(ijk), (jkl)] = (ijk) (jkl) (ijk)^{-1} (jkl)^{-1}=(il)(jk). \]

\textbf{Example 3.} In $S_{\geq 5}$,
\[ [(ijk), (klm)] = (ijk) (klm) (ijk)^{-1} (klm)^{-1} =(jkm). \]

\textbf{Example 4.} So, in fact, in $S_5$, $(123) = [(412),(253)] = [[(341),(152)],[(125),(543)]]
= [[[(234),(451)],[(315),(542)]],[[(312),(245)],[(154),(423)]]]
= [\
  [[[(123),(354)],[(245),(531)]],[[(231),(145)],[(154),(432)]]],\allowbreak
  [[[(431),(152)],[(124),(435)]],[[(215),(534)],[(142),(253)]]]\
]$.

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\textbf{Solving the Quadratic}, $ax^2+bx+c=0$: $\delta =\sqrt{\Delta}$; $\Delta =b^2-4 a c$; $r=\frac{\delta -b}{2 a}$.

\textbf{Solving the Cubic}, $ax^3+bx^2+cx+d=0$: $\Delta =27 a^2 d^2-18 a b c d+4 a c^3+4 b^3 d-b^2 c^2$; $\delta =\sqrt{\Delta }$; $\Gamma =27 a^2 d-9 a b c+3 \sqrt{3} a \delta +2 b^3$; $\gamma =\sqrt[3]{\frac{\Gamma }{2}}$; $r=-\frac{\frac{b^2-3 a c}{\gamma }+b+\gamma }{3 a}$.

\textbf{Solving the Quartic}, $ax^4+bx^3+cx^2+dx+e=0$: $\Delta _0=12 a e-3 b d+c^2$; $\Delta _1=-72 a c e+27 a d^2+27 b^2 e-9 b c d+2 c^3$; $\Delta _2=\frac{1}{27} \left(\Delta _1^2-4 \Delta _0^3\right)$; $u=\frac{8 a c-3 b^2}{8 a^2}$; $v=\frac{8 a^2 d-4 a b c+b^3}{8 a^3}$; $\delta _2=\sqrt{\Delta _2}$; $Q=\frac{1}{2} \left(3 \sqrt{3} \delta _2+\Delta _1\right)$; $q=\sqrt[3]{Q}$; $S=\frac{\frac{\Delta _0}{q}+q}{12 a}-\frac{u}{6}$; $s=\sqrt{S}$; $\Gamma =-\frac{v}{s}-4 S-2 u$; $\gamma =\sqrt{\Gamma }$; $r=-\frac{b}{4 a}+\frac{\gamma }{2}+s$.

\end{multicols}

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{\bf Theorem.} There is no general formula, using only the basic arithmetic operations and taking roots, for the solution of the quintic equation $ax^5+bx^4+cx^3+dx^2+ex+f=0$.

{\bf Key Point.} The ``persistent root'' of a closed path (path lift, in topological language) may not be closed, yet the persistent root of a commutators of closed paths is always closed.

\vskip 1mm\write\posfile{List[}

\zsavepos{proof-start}\write\posfile{List[\dimtoin{\zposy{proof-start}sp},}

{\bf Proof.} Suppose there was a formula, and consider the corresponding ``composition of machines'' picture:

\[ \import{../Sydney-1708/}{proof.pdf_t} \]

Now let $\gamma^{(1)}_1$,  $\gamma^{(1)}_2$, \ldots, $\gamma^{(1)}_{16}$,
are ``musical chairs'' paths in $X_0$ that induce permutations of the
roots and set $\gamma^{(2)}_1\coloneqq[\gamma^{(1)}_1,\gamma^{(1)}_2]$,
$\gamma^{(2)}_2\coloneqq[\gamma^{(1)}_3,\gamma^{(1)}_4]$, \ldots,
$\gamma^{(2)}_8\coloneqq[\gamma^{(1)}_{15},\gamma^{(1)}_{16}]$,
$\gamma^{(3)}_1\coloneqq[\gamma^{(2)}_1,\gamma^{(2)}_2]$, \ldots,
$\gamma^{(3)}_4\coloneqq[\gamma^{(2)}_7,\gamma^{(2)}_8]$,
$\gamma^{(4)}_1\coloneqq[\gamma^{(3)}_1,\gamma^{(3)}_2]$,
$\gamma^{(4)}_2\coloneqq[\gamma^{(3)}_3,\gamma^{(3)}_4]$, and finally
$\gamma^{(5)}\coloneqq[\gamma^{(4)}_1,\gamma^{(4)}_2]$, as in ``Dance
of the Roots'' on the next page. (Note: no ``homotopy'' anywhere). We
claim that $\gamma^{(5)}\act C\act P_1\act R_1\act\cdots\act R_4$ is a
closed path. Indeed

\parpic[r]{\includegraphics[width=0.75in]{../../Projects/Gallery/Arnold.jpg}~V.I.~Arnold\qquad}
$\bullet$ In $X_0$, none of the paths is necessarily closed.

$\bullet$ After $C$, all of the paths are closed.

$\bullet$ After $P_1$, all of the paths are still closed.

$\bullet$ After $R_1$, the $\gamma^{(1)}$'s may open up, but the $\gamma^{(2)}$'s remain closed.

$\cdots$

$\bullet$ At the end, after $R_4$, $\gamma^{(4)}$'s may open up, but $\gamma^{(5)}$ remains closed.

But if the paths are chosen as in Example 4, $\gamma^{(5)}\act C\act P_1\act R_1\act\cdots\act R_4$ is not a closed path. \hfill $\Box$

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{\bf References.}
V.I. Arnold, 1960s, hard to locate.

V.B. Alekseev, {\em Abel's Theorem in Problems and Solutions, Based on the Lecture of Professor V.I. Arnold,} Kluwer 2004.

A. Khovanskii, {\em Topological Galois Theory, Solvability and Unsolvability of Equations in Finite Terms,} Springer 2014.

B. Katz, {\em Short Proof of Abel's Theorem that 5th Degree Polynomial Equations Cannot be Solved,} YouTube video, \url{http://youtu.be/RhpVSV6iCko}.

\newpage

\null\hfill{\bf Dance of the Roots.}

\vskip 1mm

\zsavepos{dance-start}\write\posfile{List[\dimtoin{\zposy{dance-start}sp},}

{
\def\g#1#2{$\gamma^{(1)}_{#1}\sim(#2)$}
\def\a{$\gamma^{(2)}_1=[\gamma^{(1)}_1,\gamma^{(1)}_2]=$}
\def\b{$\gamma^{(2)}_2=[\gamma^{(1)}_3,\gamma^{(1)}_4]=$}
\def\c{$\gamma^{(2)}_3=[\gamma^{(1)}_5,\gamma^{(1)}_6]=$}
\def\d{$\gamma^{(2)}_4=[\gamma^{(1)}_7,\gamma^{(1)}_8]=$}
\def\e{$\gamma^{(3)}_1=[\gamma^{(2)}_1,\gamma^{(2)}_2]=$}
\def\f{$\gamma^{(3)}_2=[\gamma^{(2)}_3,\gamma^{(2)}_4]=$}
\def\h{$\gamma^{(4)}_1=[\gamma^{(3)}_1,\gamma^{(3)}_2]=$}
\[ \resizebox{0.92\textwidth}{!}{\input{Dance.pdftex_t}} \]
}

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\null\hfill{\bf Yes, Prime Minister, 1986.}

\begin{multicols}{2}
{\bf Sir Humphrey:} You know what happens: nice young lady comes up to you. Obviously you want to create a good impression, you don't want to look a fool, do you? So she starts asking you some questions: Mr. Woolley, are you worried about the number of young people without jobs?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} Are you worried about the rise in crime among teenagers?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} Do you think there is a lack of discipline in our Comprehensive schools?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} Do you think young people welcome some authority and leadership in their lives?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} Do you think they respond to a challenge?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} Would you be in favour of reintroducing National Service?
\newline{\bf Bernard Woolley:} Oh...well, I suppose I might be.
\newline{\bf Sir Humphrey:} Yes or no?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} Of course you would, Bernard. After all you told me can't say no to that. So they don't mention the first five questions and they publish the last one.
\newline{\bf Bernard Woolley:} Is that really what they do?
\newline{\bf Sir Humphrey:} Well, not the reputable ones no, but there aren't many of those. So alternatively the young lady can get the opposite result.
\newline{\bf Bernard Woolley:} How?
\newline{\bf Sir Humphrey:} Mr. Woolley, are you worried about the danger of war?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} Are you worried about the growth of armaments?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} Do you think there is a danger in giving young people guns and teaching them how to kill?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} Do you think it is wrong to force people to take up arms against their will?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} Would you oppose the reintroduction of National Service?
\newline{\bf Bernard Woolley:} Yes
\newline{\bf Sir Humphrey:} There you are, you see Bernard. The perfect balanced sample.
\end{multicols}

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