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\begin{document}

\begin{center}
{\Large Advanced Algebra I}\\
 Homework 9\\
 due on Dec. 8, 2006
\end{center}


\begin{enumerate}
\item Complete the uncompleted proof in the lecture.

\item Let $k[u_1,...,u_n]$ be a polynomial ring with $n$
indeterminates. Let $s_i:= \sum_{j_1 < j_2 <¡K<j_i}
u_{j_1}u_{j_2}...u_{j_i}$ be the elementary symmetric polynomials.
Show that $f \in k[u_1,...,u_n]$ is symmetric if and only if $f
\in k[s_1,...,s_n]$.

In fact, one can consider the group action $S_n$ on
$k[u_1,...,u_n]$. Let $k[u_1,...,u_n]^{S_n}$ be the {\bf
invariant}, i.e. $ \{f| \sigma(f) = f, \forall \sigma \in S_n\}$.
The above assertion can be rephrased as $ k[u_1,...,u_n]^{S_n}
=k[s_1,...,s_n]$.

One can show that $k(u_1,...,u_n)^{S_n} = k(s_1,...,s_n)$.

\item Keep the notation as above. Determine the Galois group of
$k(u_1,...,u_n)$ over $k(s_1,...,s_n)$.

\item Show that for any given finite group $G$. There exists a
Galois extension $F/K$
with  Galois group $G$. \\
\noindent (Remark: It would be a very difficult problem (Inverse
Galois Problem) if the base field is fixed, e.g $\bQ$.)

\item Determine the Galois group of $x^3+x+1$ over $\bQ$ and over
$\mathbb{F}_5$ respectively.

\item Determine the Galois group of $x^4+x+1$ over $\bQ$ and over
$\mathbb{F}_7$ respectively.

\item Determine the Galois group of $x^7-3$ over $\bQ$.

\item If $f(x)= x^4+bx^3+cx^2+dx+e$, then its resolvant cubic is
$g(x)= x^3-c x^2+ (bd-4e)x-b^2e+4ce-d^2$.
\end{enumerate}

\end{document}
\end