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\title{Math 728\\Fall 2004\\Homework 1}
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\centerline{\it Due Wednesday September 1, 2004.}

\centerline{}

\noindent 1. (5 pt) Give an example of a division ring that is {\it not} a field.

\centerline{}

\noindent 2. Consider the set $\mathfrak{C}=\{f\vert f:\mathbb{R}\longrightarrow\mathbb{R}\}$ of functions from $\mathbb{R}$ to $\mathbb{R}$. Also consider the collection of functions $X=\{f_{\alpha}\vert\alpha\in\mathbb{R}\}$ where $f_{\alpha}(x)$ is defined by

\[
f_{\alpha}(x)=
\begin{cases}
	1 & \text{ if $x=\alpha$},\\
	0 & \text{ if $x\neq\alpha$}.
\end{cases}
\]




\begin{itemize}
\item[a)] (5 pt) Show that $\mathfrak{C}$ is a real vector space.
\item[b)] (5 pt) Show that $X$ is a linearly independent subset of $\mathfrak{C}$.
\item[c)] (5 pt) If $X$ a basis for $\mathfrak{C}$ over $\mathbb{R}$? If not can you construct a basis for $\mathfrak{C}$ over $\mathbb{R}$ containing $X$?
\end{itemize}

\centerline{}

\noindent 3. Let $\mathbb{K}\subseteq\mathbb{F}$ be fields.
\begin{itemize}
\item[a)] (5 pt) Show that $\mathbb{F}$ is a vector space over $\mathbb{K}$.
\item[b)] (5 pt) Show that if $\text{dim}_{\mathbb{K}}\mathbb{F}=n<\infty$ then every element of $\mathbb{F}$ is algebraic over $\mathbb{K}$.
\item[c)] (5 pt) Give an example of an infinite extension of fields that is algebraic and an example of an infinite extension of fields that is not algebraic.
\end{itemize}



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