\documentclass[12pt]{amsart} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage[all]{xy} \voffset=-1in \hoffset=-1in \textheight=10in \textwidth=6in \begin{document} \author{} \title{Math 728\\Fall 2004\\Homework 1} \date{} \maketitle \thispagestyle{empty} \pagestyle{empty} \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} \end{document}