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\def\front{{\parbox[t]{3.8in}{
{\scriptsize
  Dror Bar-Natan: Classes: 2007-08: Math 401 Polynomials, Equations, Fields:
}
\begin{center}
  \large\bf\color{red}Galois Theory Quick Reference
\end{center}
\par\noindent{\bf\color{red}Goal.} Some polynomials cannot be ``solved''
using $+$, $-$, $\times$, $\div$ and $\sqrt[n]{\quad}$.
\par\noindent{\bf\color{red}Galois Theory.} Roughly, there is a correspondence
\begin{center} \begin{tabular}{|ccc|}
  \hline
  \{field extensions\} &
  $\overset{\text{\tiny The Fundamental}}
    {\underset{\text{\tiny Theorem}}{
      \longleftrightarrow
  }}$ &
  \{groups\} \\
  \hline
  \{extensions by roots\} &
  $\longrightarrow$ &
  \{``solvable groups''\} \\
  \hline
  \parbox{1.2in}{\centering splitting field of $3x^5-15x+5$} &
  $\longrightarrow$ &
  \parbox{1.2in}{\centering the non-solvable permutation group $S_5$} \\
  \hline
\end{tabular} \end{center}
\par\noindent{\bf\color{red}To do.} 
\begin{enumerate}
\item More on splitting fields.
\item Quick reminders on group theory.
\item Precise statement of the fundamental theorem.
\item Examples for the fundamental theorem.
\item On solvable groups: definition, basic properties, $S_5$ is not
  solvable.
\item ``Extensions by radicals'' correspond to solvable groups.
\item The splitting field of $3x^5-15x+5$ corresponds to $S_5$.
\item Proof of the fundamental theorem.
\end{enumerate}
}}}

\def\back{{\parbox[t]{3.8in}{
{\bf\color{red}The Fundamental Theorem of Galois Theory.} Let $F$ be a
field of characteristic 0 and let $E$
be a splitting field over $F$.  Then there is a bijective correspondence
between the set $\{K:E/K/F\}$ of intermediate field extensions $K$
lying between $F$ and $E$ and the set $\{H:H<\operatorname{Gal}(E/F)\}$
of subgroups $H$ of the Galois group $\operatorname{Gal}(E/F)$ of the
original extension $E/F$:
\vskip -1mm
\[ \{K:E/K/F\}\quad\leftrightarrow\quad\{H:H<\operatorname{Gal}(E/F)\}.\]
\vskip -1mm
The bijection is given by mapping every intermediate extension
$K$ to the subgroup $\operatorname{Gal}(E/K)$ of
elements in $\operatorname{Gal}(E/F)$ that preserve $K$,
\vskip -1mm
\[ \Phi:\ K\mapsto\operatorname{Gal}(E/K):=\{g:E\to E:g|_K=I\}, \]
\vskip -1mm
and reversely, by mapping every subgroup $H$ of
$\operatorname{Gal}(E/F)$ to its fixed field $E_H$:
\vskip -1mm
\[ \Psi:\ H\mapsto E_H:=\{x\in E:\forall h\in H,\ hx=x\}. \]
\vskip -1mm
This correspondence has the following further properties:
\begin{itemize}
\item  It is inclusion-reversing: if $H_1\subset H_2$ then
$E_{H_1}\supset E_{H_2}$ and if $K_1\subset K_2$ then
$\operatorname{Gal}(E/K_1)>\operatorname{Gal}(E/K_2)$.
\item  It is degree/index respecting:
$[E:K]=|\operatorname{Gal}(E/K)|$ and
$[K:F]=[\operatorname{Gal}(E/F):\operatorname{Gal}(E/K)]$.
\item  Splitting fields correspond to normal subgroups: If $K$ in
$E/K/F$ is the splitting field of a polynomial in
$F[x]$ then $\operatorname{Gal}(E/K)$ is normal in
$\operatorname{Gal}(E/F)$ and
$\operatorname{Gal}(K/F)\cong\operatorname{Gal}(E/F)/\operatorname{Gal}(E/K)$.
\end{itemize}
}}}

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