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\centerline{Mathematics 352 -- Abstract Algebra II}
\centerline{Discussion 2 -- More on Galois Groups}
\centerline{March 28, 2008}
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\noindent
{\it Background}
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Given an extension field $K$ of a field $F$, recall that we have defined
\[
{\rm Gal}_F K = \{\sigma : K\to K : \sigma \text{ is an $F$-automorphism of $K$. }\}
\]
which is a group under the operation of function composition, called the 
\emph{Galois group} of $K$ over $F$.  For instance, last Friday we saw that
${\rm Gal}_\Q \Q(\sqrt{2},\sqrt{3}) = \{id, \rho, \sigma, \tau\}$ is a noncyclic group
of order $4$, isomorphic to $(\Z_2 \times \Z_2, +)$.  Today we want to work out 
another example and start to understand a relationship between subgroups of 
${\rm Gal}_F K$ and subfields of $K$ containing $F$.  
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\noindent
{\it Discussion Questions}
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\noindent
A)  Consider the polynomial $f(x) = x^3 - 2 \in \Q[x]$, and 
let $K$ be the splitting field of $f(x)$ over $\Q$.  
\begin{enumerate}
\item[(1)]  Show that $K = \Q({\root 3\of 2}, \omega)$, where 
$\displaystyle{\omega = \frac{-1 + i\sqrt{3}}{2}}$ is a cube root of $1$.  
\item[(2)] Consider the minimal polynomials $f(x) = x^3 - 2$ of ${\root 3\of 2}$
and $g(x) = x^2 + x + 1$ of $\omega$ in $\Q[x]$.
By the theorem we finished proving in class last
Friday, we know that given any root $v$ of $f(x)$ and any root $w$ of $g(x)$, 
there must be some element $\phi$ of ${\rm Gal}_\Q \Q({\root 3\of 2},\omega)$ 
with $\phi({\root 3\of 2}) = v$
and $\phi(\omega) = w$.  How many distinct elements of the 
${\rm Gal}_\Q \Q({\root 3\of 2},\omega)$ are there?  List them all.
\item[(3)]  By numbering the 3 roots of $x^3 - 2$ in some way, establish
a 1-1 correspondence between ${\rm Gal}_\Q \Q({\root 3\of 2},\omega)$ 
and a subgroup of $S_3$.  (Which permutation of the roots is induced by 
each element of the Galois group?)
What is the structure of ${\rm Gal}_\Q \Q({\root 3\of 2},\omega)$?
(i.e. to which ``standard'' group is it isomorphic?)
\end{enumerate}
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\noindent
B)  Now we want to consider subgroups of ${\rm Gal}_F K$ and subfields of $K$.
\begin{enumerate}
\item[(1)] Show in general that if $G = {\rm Gal}_F K$ and $H$ is a subgroup 
of $G$, then 
\[
E_H = \{a \in K : \sigma(a) = a \text{ for all } \sigma \in H\}
\]
is a \emph{subfield} of $K$ containing $F$ (called the \emph{fixed field} of 
$H$ for a hopefully clear reason!).  Note that this requires you to show
that $E_H$ is a subring of $K$ which is also a field.  
\item[(2)] Let $F = \Q$ and $K = \Q(\sqrt{2},\sqrt{3})$.  The Galois group
$G = {\rm Gal}_\Q \Q(\sqrt{2},\sqrt{3})$ has 5 different subgroups $H$ (including 
the ``trivial'' subgroups $\{id\}$ and $G$).  What are the fixed fields
of each of them?
\item[(3)] The Galois group $G = {\rm Gal}_\Q \Q({\root 3\of 2},\omega)$
has 6 different subgroups $H$ (again including the trivial subgroups). 
Find the fixed fields of each one.  
\end{enumerate}
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\noindent
{\it Assignment}
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Group writeups due in class on Monday, April 7.
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