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\begin{document}

\begin{center}
{\bf\large  ALGEBRA TEST \#III}
\end{center}

\vspace{1in}

This exam is due Wednesday, May 9.
You are expected to complete three problems,
one from each group.
Clearly mark which problems are to be graded.

You may use your book, but you may not
communicate with others concerning
the exam.
In order to receive full credit your answer must be 
{\bf complete}, {\bf legible} and {\bf correct}. 

\vfill

\begin{center}
I have neither given nor received aid on this exam.
\end{center}

\vspace{.5in}

\begin{flushright}
{\bf Name:\underline{\hspace{2.5in}}}
\end{flushright}


\break

\noindent
{\bf Group 1.} (There is only one problem in Group 1.)

\medskip

1. True or False? (Justify your answer with 
a brief proof or counterexample.)
\begin{enumerate}
\item[(i)] Every field extension of degree $2$ is Galois.
\item[(ii)] ${\BQ}<(\overline{{\BQ}}\cap{\BR})$ is Galois.
\item[(iii)] ${\rm Gal}_{\BQ}(x^3-x+1)\cong S_3$.
\item[(iv)] If $\alpha = \sqrt[5]{2+i} + \sqrt[5]{2-i}$,
then ${\rm Gal}({\BQ}[\alpha]/{\BQ}) \cong S_5$.
\item[(v)] If $f(x)\in {\BF}[x]$ is irreducible and has
degree $n$, then ${\rm Gal}_{\BF}(f)$ has an element
of order $n$.
\end{enumerate}

\vfill

\noindent
{\bf Group 2.}

\medskip

2. Show that if ${\BE}$ and ${\BK}$ are finite 
Galois extensions of ${\BF}$, and ${\BF} = {\BE}\cap {\BK}$, then 
$[{\BE}{\BK}:{\BF}] = [{\BE}:{\BF}] [{\BK}:{\BF}]$.

\vfill

3.  Let $p$ be an odd prime and let
$\zeta_1 = \omega, \zeta_2 = \omega^2,\ldots, \zeta_{p-1}
= \omega^{p-1}$ be the 
different primitive $p$-th roots of unity.
\begin{enumerate}
\item[(a)] Show that $u_k = 
\zeta_1^k + \cdots + \zeta_{p-1}^k$ equals
$p-1$ if $p\mid k$ and equals $-1$ otherwise.
\item[(b)] Show that the discriminant of 
$\Phi_{p}(x) = x^{p-1} + \cdots + x + 1$ is 
$(-1)^{\frac{p-1}{2}}p^{p-2}$.
\end{enumerate}

\vfill

\noindent
{\bf Group 3.}

\medskip

4. Show that an algebraic 
extension of a perfect field is perfect.

\vfill

5. Let $p$ be an odd prime. Describe the Galois
group of $x^p-2$ over ${\BQ}$.

\vfill

\end{document}
