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\parbox[b]{2in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}: \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \newline \href{http://www.math.toronto.edu/~drorbn/classes/\#2526}{2025-26}: \href{http://drorbn.net/25-347}{MAT 347 Groups, Rings, Fields}:
}~{\LARGE\bf Galois Summary}
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\vspace{-4mm}

\begin{multicols}{2}

\noindent {\bf\large Selick.}

\begin{enumerate}

\item {\bf Thm 3.1.1.} The $\gcd$ in an extension field is the same as in the base field.

\item {\bf Thm 3.1.4.} Eisenstein's Criterion). $R$ a UFD, $p\in R$ a prime, $f=\sum_0^n a_ix^i$ with $p||a_0$, $p|a_1$, \ldots, $p|a_{n-1}$, $p\nmid a_n$. Then $f$ is irreducible in $Q(R)[x]$.

\item {\bf Prop 3.2.2.} $[L:F] = [L:K][K:F]$.

\item {\bf Lem 3.2.8.} The minimal polynomial is irreducible.

\item {\bf Lem 3.2.9.} If $a$ is algebraic over $F$, then $[F(a):F]$ is the degree of the minimal polynomial of $a$.

\item {\bf Lem 3.2.10.} Also, $F(a) = F[a]$.

\item {\bf Cor 3.2.11.} Also, $F(a) \cong F[x]/(\text{min.\ poly.\ of }a)$.

\item {\bf Thm 3.2.12.} The set of algebraic elements over $F$ is a field.

\item {\bf Cor 3.2.13.} If $L/K/F$ and $K/F$ and $L/K$ are algebraic, then so is $L/F$.

\item {\bf Def 3.3.9.} A splitting field.

\item {\bf Thm 3.3.13.} If $E/F$, $E'/F'$, $\tau\colon F\to F'$ a morphism, $p\in F[x]$ irreducible, $p'=\tau p$, $a$, $a'$ roots of $p$, $p'$ in $E$, $E'$, then there is an extension $\tilde{\tau}\colon E\to E'$ of $\tau$ such that $\tilde{\tau}(a)=a'$. (``all roots of an irreducible polynomial are the same'').

\item {\bf Thm 3.3.15.} If $\tau\colon F\to F'$ an isomorphism, $p\in F[x]$ irreducible, $p'=\tau p$, $E$ is a splitting field of $p$ and $E'$ is a splitting field of $p'$, then there is an extension of $\tau$ to an isomorphism $\tilde{\tau}\colon E\to E'$.

\item {\bf Thm 3.4.1.} Frobenius: If $\chara F=p$ then $(a+b)^p=a^p+b^p$ and more generally, $(a+b)^{p^k}=a^{p^k}+b^{p^k}$.

\item {\bf Thm 3.5.1.} $f$ has repeated roots iff $\gcd(f,f')\neq 1$.

\item {\bf Cor 3.5.2.} $f\in F[x]$ irreducible. If $f$ has repeated roots then $p\coloneqq\chara f\neq 0$ and $f(x)=g(x^p)$ for some $g$.

\item {\bf Cor 3.6.2.} A finite field has $q=p^n$ elements.

\item {\bf Cor 3.6.3} (Fermat). In a finite field with $q$ elements, $\forall a\, a^q=1$.

\item {\bf Thm 3.6.4.} A finite field with $q$ elements is a splitting field of $x^q-x$ over $\bbF_p$ (and thus any two are isomorphic).

\item {\bf Thm 3.6.7.} The roots of $x^q-x$ in its splitting field make a field, and hence $\bbF_q$ exists.

\item {\bf Thm 3.6.8.} In a finite group, if for every $n$ there are at most $n$ elements with $g^n=e$, then $G$ is cyclic.

\item {\bf Cor 3.6.9.} Any finite subgroup of the multiplicative group of a field is cyclic.

\item {\bf Cor 3.6.10.} The multiplicative group of $\bbF_q$ is cyclic.

\item {\bf Def 3.7.1.} Separable elements, separable extensions.

\item {\bf Prop 3.7.2.} in characteristic 0, every extension is separable.

\item {\bf Example 3.7.3.} If $\chara F=p$ and $E=F(z)$ then $E(z^{1/p})/E$ is not separable.

\item {\bf Thm 3.7.4.} A separable extension can be generated by a single element.

\item {\bf Thm 3.8.2.} Distinct field automorphisms are linearly independent.

\item {\bf Thm 3.8.3.} The fixed set $E^S$ of a set $S$ of automorphisms of $F$ is a field, ``the fixed field''.

\item {\bf Def.} $\Gal(E/F)$.

\item {\bf Claim} $E^{\Gal(E/F)} \supset F$.

\item {\bf Thm 3.8.5.} $|\Gal(E/F)|\leq[E:F]$.

\end{enumerate}

\vspace{-3mm}\noindent\rule{\linewidth}{1pt}

\noindent {\bf\large Gallian.}

\begin{enumerate}

\item {\bf Page 557.} $\Gal(3x^5-15x+5) \cong S_5$.

\end{enumerate}

\vspace{-3mm}\noindent\rule{\linewidth}{1pt}

\noindent {\bf\large 08-401 handout ``Fundamental Theorem''.}
Assuming $\chara F=0$.

\begin{enumerate}

\item {\bf The Fundamental Theorem.} Given a splitting $E/F$, there is a bijection $\{K\colon E/K/F\} \longleftrightarrow \{H\colon H<\Gal(E/F)\}$ which is
    \begin{enumerate}[left=0pt .. 24pt]
    \item Inclusion reversing.
    \item Degree/index respecting: $[E:K]=|\Gal(E/K)|$ and $[K:F]=(\Gal(E/F):\Gal(E/K))$.
    \item Splitting fields correspond to normal subgroups: If $K$ in $E/K/F$ is the splitting field of a polynomial in $F[x]$ then $\Gal(E/K)$ is normal in $\Gal(E/F)$ and $\Gal(K/F)\cong\Gal(E/F)/\Gal(E/K)$.
    \end{enumerate}

\end{enumerate}

\end{multicols}

\end{document}


