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{\bf Test 2 for Honors Algebra IV Math 362 \\ }
{\footnotesize \hfill April 3, 1998}\\
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Instructions: The test is 60 minutes in length.
\begin{enumerate}
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\item (20 pts) Assume $\F$ is a subfield of the complex numbers,
i.e. $\F\subset \C$. Assume $\sigma_1,\sigma_2,\sigma_3$ are
three monomorphisms from $ \F\rightarrow \C$. Assume we have
the identity $\sigma_3(x)=2\sigma_2(x)-\sigma_1(x)$ for all
$x\in\F$. What can you say about these three monomorphisms.
Explain.
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\item (10 pts) Write down an irreducible polynomial which is not
separable. Explain why your example has the stated properties.
\vfill
(10 pts) Write down an irreducible polynomial which is not
normal. Explain why your example has the stated properties.
\vfill
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\item (10 pts) Describe in detail the splitting field of the
polynomial $x^5-1\in\Q[x]$.
\vfill
(10 pts) Compute the Galois group and write down the Galois
correspondence indicating on one side the subgroups of the
Galois group and on the other side the corresponding subfields.
\vfill
\newpage
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\item (10 pts) Compute the normal closure $N$ of the field
extension $\Q\left(\sqrt[3]{2}\right)\supset\Q$ inside the
complex numbers $\C$.
\vfill
(10 pts) Compute the Galois group $\Gamma[N:\Q]$. Hint: You
either can describe the Galois group in an explicit way or you
can find it by some general arguments.
\vfill
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\item (20 pts) Let $GF(1024)$ be the finite field of $1024$
elements. Describe explicitly the automorphisms of the Galois
group $\Gamma [GF(1024):{\mathbb Z}_2]$. Then work out the
Galois correspondence indicating on one side the subgroups of
$\Gamma$ and on the other side the subfields of $GF(1024)$.
\vfill
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\end{enumerate}
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