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\begin{document}

\parindent 0pt
\parbox[b]{2in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}:
  \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \href{http://www.math.toronto.edu/~drorbn/classes/\#2526}{2025-26}:
  \newline \href{http://drorbn.net/25-347}{MAT 347 Groups, Rings, Fields}:
}~{\LARGE\bf Monsters}
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\vskip -3mm
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\noindent

\textbf{Solving the Linear}, $ax+b=0$: $x=-b/a$:

\vskip 2mm
\includegraphics[scale=0.72]{M1.pdf}

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

\vskip 2mm
\includegraphics[scale=0.72]{M2.pdf}

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

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\includegraphics[scale=0.72]{M3.pdf}

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

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\includegraphics[scale=0.72]{M4.pdf}

%\[ \scalebox{0.333}{\parbox{3\linewidth}{$
%x= -\frac{1}{2} \sqrt{\frac{b^2}{4 a^2}+\frac{\sqrt[3]{\sqrt{\left(-72 a c e+27 a d^2+27 b^2 e-9 b c d+2 c^3\right)^2-4 \left(12 a e-3 b
%   d+c^2\right)^3}-72 a c e+27 a d^2+27 b^2 e-9 b c d+2 c^3}}{3 \sqrt[3]{2} a}+\frac{\sqrt[3]{2} \left(12 a e-3 b d+c^2\right)}{3 a \sqrt[3]{\sqrt{\left(-72 a
%   c e+27 a d^2+27 b^2 e-9 b c d+2 c^3\right)^2-4 \left(12 a e-3 b d+c^2\right)^3}-72 a c e+27 a d^2+27 b^2 e-9 b c d+2 c^3}}-\frac{2 c}{3 a}}-\frac{b}{4 a}$
%  \newline \vskip5mm
%  $-\frac{1}{2}
%   \sqrt{\frac{b^2}{2 a^2}-\frac{-\frac{b^3}{a^3}+\frac{4 b c}{a^2}-\frac{8 d}{a}}{4 \sqrt{\frac{b^2}{4 a^2}+\frac{\sqrt[3]{\sqrt{\left(-72 a c e+27 a d^2+27
%   b^2 e-9 b c d+2 c^3\right)^2-4 \left(12 a e-3 b d+c^2\right)^3}-72 a c e+27 a d^2+27 b^2 e-9 b c d+2 c^3}}{3 \sqrt[3]{2} a}+\frac{\sqrt[3]{2} \left(12 a
%   e-3 b d+c^2\right)}{3 a \sqrt[3]{\sqrt{\left(-72 a c e+27 a d^2+27 b^2 e-9 b c d+2 c^3\right)^2-4 \left(12 a e-3 b d+c^2\right)^3}-72 a c e+27 a d^2+27 b^2
%   e-9 b c d+2 c^3}}-\frac{2 c}{3 a}}}-\frac{\sqrt[3]{\sqrt{\left(-72 a c e+27 a d^2+27 b^2 e-9 b c d+2 c^3\right)^2-4 \left(12 a e-3 b d+c^2\right)^3}-72 a c
%   e+27 a d^2+27 b^2 e-9 b c d+2 c^3}}{3 \sqrt[3]{2} a}-\frac{\sqrt[3]{2} \left(12 a e-3 b d+c^2\right)}{3 a \sqrt[3]{\sqrt{\left(-72 a c e+27 a d^2+27 b^2
%   e-9 b c d+2 c^3\right)^2-4 \left(12 a e-3 b d+c^2\right)^3}-72 a c e+27 a d^2+27 b^2 e-9 b c d+2 c^3}}-\frac{4 c}{3 a}}
%$}} \]

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\textbf{Solving the Quintic}, $ax^5+bx^4+cx^3+dx^2+ex+f=0$: An even bigger monster? Galois: Not in this universe.

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