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\title{Math 772\\Summer 2006\\Homework 5}
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\centerline{\it Due Wednesday, August 2, 2006 (HA HA).}

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\noindent 1. (5 pt) Show that if $\mathbb{F}$ is a field such that $\text{char}(\mathbb{F})\neq 2$, then there is a one to one correspondence between the set of quadratic extensions of $\mathbb{F}$ and the nontrivial elements of the group $\mathbb{F}^*/(\mathbb{F}^*)^2$.

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\noindent 2. For the following list of fields, find the number of quadratic extensions.
\begin{itemize}
\item[a)] (5 pt) $K$ where $K$ is any algebraic number field.
\item[b)] (5 pt) $\mathbb{F}$ where $\mathbb{F}$ is any finite field.
\item[c)] (5 pt) $\mathbb{R}$.
\item[d)] (5 pt) $\mathbb{C}$.
\item[e)] (5 pt) $\mathbb{Q}_p$ where $p$ is an odd prime.
\item[f)] (5 pt) $\mathbb{Q}_2$.
\end{itemize}

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\noindent 3. (5 pt) Is there a countable field of characteristic $0$ that possesses a unique quadratic extension (if so, give an example and if not prove that one cannot exist)?

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\noindent 4. (5 pt) Show that the integers $\mathbb{Z}$ form a dense subset (with respect to the $p-$adic metric) of $\mathbb{Z}_p$. What is $\overline{\mathbb{Z}}\bigcap\mathbb{Q}$ (where $\overline{\mathbb{Z}}$ is the closure of $\mathbb{Z}$ with respect to the $p-$adic metric)?

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{\it My summer begins when this homework ends...}-C. Hashbarger

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