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{\bf Test 1 for Honors Algebra IV Math 362 \\ }
{\footnotesize \hfill March 2, 1998}\\
\end{center}
Instructions: The test is 60 minutes in length.
\begin{enumerate}
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\item (15 pts) Assume the points $(0,0)$ and $(1,0)$ are given in
the plane. Use ruler and compass to construct the regular
square having vertices at $(0,0)$, $(0,\sqrt{2})$,
$(\sqrt{2},0)$ and $(\sqrt{2},\sqrt{2})$. Provide details for
your constructions.
\vfill
\hrulefill
Construction steps:\vspace{3.5mm}\\
1.\vspace{3.5mm}\\
2.\vspace{3.5mm}\\
3.\vspace{3.5mm}\\
4.\vspace{3.5mm}\\
5.\vspace{3.5mm}\\
6.\vspace{3.5mm}\\
7.\vspace{3.5mm}\\
8.\vspace{3.5mm}\\
9.
\newpage
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\item (8 pts) Construct the splitting field ${\mathbb F}$ of the
polynomial $x^4-4\in {\mathbb Q}[x]$ inside the complex field
${\mathbb C}$.
\vfill
(7 pts) Compute and describe the Galois group $\Gamma [{\mathbb
F}:{\mathbb Q}]$.
\vfill
\newpage
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\item (15 pts) Let $\F=\Z_{11}$ be the field of 11 elements.
Consider the set of all vectors $x\in\F^{10}$ satisfying
$$
\sum_{i=0}^{10}ix_i=0 \mbox{ and } \sum_{i=0}^{10}x_i=0.
$$
Compute the cardinality of the set described above.
\vfill
(5 pts bonus) Show that above `special ISBN numbers' can
correct one error. This means: If at most one of the digits
$x_i$ was wrongly recorded as a digit $\tilde{x}_i\neq x_i$
then it is possible to compute the vector $x=(x_1,\ldots,x_i,
\ldots, x_{10})$ from the knowledge of the vector
$(x_1,\ldots,\tilde{x}_i, \ldots, x_{10})$.
\vfill
\newpage
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\item (15 pts) If $u\in \F$ is algebraic of odd degree over $\K$,
then so is $u^2$ and $\K(u)=\K(u^2)$.
\newpage
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\item (20 pts) If $f\in \K[x]$ has degree $n$ and $f$ is a
splitting field of $f$ over $\K$ then $[\F:\K]$ divides $n!$.
\newpage
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\item (5 pts) Construct a field $\F$ of 8 elements.\vfill
(5 pts) Show that every field of 8 elements must have $\Z_2$ as
a prime field.\vfill
(5 pts) Determine all intermediate fields of $[\F:\Z_2]$.\vfill
(5 pts) Describe a nontrivial field automorphism
$\varphi:\F\longrightarrow\F$.\vfill
(5 pts bonus) Compute the Galois group $\Gamma [{\mathbb
F}:{\mathbb Z}_2]$.\vfill
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\end{enumerate}
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