\documentclass[11pt]{article}
\usepackage{fullpage}
\usepackage{amssymb,amsfonts,amsmath,amsthm} 
\usepackage[margin= 1in]{geometry}
\usepackage{url}

\usepackage{enumerate}
\usepackage{color}
\usepackage[pdftex]{graphicx}
\usepackage{lastpage}

\newtheorem{thm}{THEOREM}[]
\newtheorem{lem}[thm]{LEMMA}
\newtheorem{cor}[thm]{COROLLARY}
\newtheorem{prop}[thm]{PROPOSITION}
\newtheorem{as}[thm]{ASSUMPTION}
\newtheorem{conjecture}[thm]{Conjecture}
\newtheorem{claim}[thm]{CLAIM}
\theoremstyle{definition}
\newtheorem{defi}[]{DEFINITION}
\theoremstyle{definition}
\newtheorem{remark}[thm]{REMARK}
\newtheorem{fact}[thm]{Fact}
\newtheorem{ex}[thm]{Example}

\setlength{\parindent}{0cm}

\newcommand{\tab}{\hspace{3ex}}

\usepackage{titlesec}
\usepackage{sectsty}
\hyphenchar\font=-1

\sectionfont{  \normalsize	\sectionrule{0pt}{0pt}{-5pt}{0.05pt}
	}

\titleformat*{\section}{ \Large  \center}
\titleformat{\subsection}[runin]
               {\normalfont\bfseries }
               {}{0em}{}[.]
               
\titlespacing{\section}
               {\parindent}{0pt}{5pt}
\titlespacing{\subsection}
               {0pt}{2ex plus .1ex minus .2ex}{5pt} 
               
               
               \makeatletter
 \renewcommand{\@evenfoot}{%
   Page \thepage{} of \pageref{VeryLastPage} \hfil
    Math CS117, \copyright Katy Craig, 2025%
 }
 \renewcommand{\@oddfoot}{\@evenfoot}
 \makeatother

               
\begin{document}

\begin{center}
{\huge  \sc  Math CS 117: Homework 1} \\
{\large  Due  Monday, April 7th at 11:59pm}
\end{center}
Questions followed by * are to be turned in. Questions without * are extra practice. At least one extra practice question will appear on each exam.

 \section*{Question 1 }
 Suppose $F$ is a field and $a,b,c \in F$. Prove the following:
 \begin{enumerate}[(a)]
 \item $(-a)b =-ab$;
 \item $(-a)(-b) = ab$;
 \item $ac = bc$ and $c \neq 0$ implies $a =b$;
 \item $ab=0$ implies either $a=0$ or $b=0$.
 \end{enumerate}
 
 \section*{Question 2*}
 Suppose $F$ is an ordered field and $a,b, \epsilon \in F$. Recall that, if 1 is the multiplicative identity of $F$, we define $2:=1+1$. Prove the following:
 \begin{enumerate}[(a)]
 \item if $0< a< b$, then $0 < 1/b < 1/a$;
 \item $2ab \leq a^2 + b^2$;
 \item if $a \leq b+\epsilon$ for all   $\epsilon >0$, then $a \leq b$.
 \end{enumerate}
 \textbf{We will use this last property repeatedly throughout the course.}
 
 \section*{Question 3}
In the definition of a field, suppose that the condition $1\neq 0$ in item M4 was removed. Prove that, if $F$ is a field for which $1 = 0$, then $F  = \{0\}$.

\vspace{1cm}
\hrulefill \\
\textbf{Note}: From now on, you do not need to cite which individual properties of an ordered field (e.g. (A1), (A2), etc.) that you are using, and you may combine a few properties in a single logical step. \

\hrulefill \\
\bigskip

\section*{Question 4*}
Consider the \emph{Gaussian rational field} $\mathbb{Q}(i)$, defined to be the set
\[ \mathbb{Q}(i) = \{ p + q i : p, q \in \mathbb{Q} \} , \]
where $i$ denotes an element satisfying $i^2 = -1$. As the name implies, the Gaussian rational field is a field, endowed with the following addition and multiplication operations:
\[ (p+qi) + (p'+q'i) = (p+p') + (q+q')i , \quad (p+qi)\cdot (p'+q'i)  = (pp' - qq') + (pq'+qp')i . \]
The additive identity is $0 = 0 + 0i$ and the multiplicative identity is $1 = 1+0i$. 

\bigskip\bigskip

We may endow the Gaussian rational field with the \emph{lexicographical ordering} given by 
\[ p+qi \leq p' + q'i \iff \text{ either (i) } p < p' \text{ or (ii) } p = p' \text{ and } q  \leq q' .\]
(This is sometimes known as the \emph{dictionary ordering}, since it follows the same principle by which one puts a list of words in alphabetical order.)
\begin{enumerate}[(a)]
\item For any $x,y \in \mathbb{Q}(i)$ prove that exactly one of the following is true: $x< y$, $x=y$ or $x>y$.
\item For any $x,y,z \in \mathbb{Q}(i)$, if $x\leq y$ and $y\leq z$, prove that $x\leq z$.
\item Even though $\mathbb{Q}(i)$ is a field and can be endowed with an ordering as described above, it is \emph{not} an ordered field. Which part of the definition of an ordered field is violated? Justify your answer.
\end{enumerate}

 

\section*{Question 5}
Suppose $F$ is an ordered field and $a,b \in F$. Prove the following basic properties of the absolute value stated in class.
\begin{enumerate}[(a)]
\item $|a| \geq 0$
\item $|ab| = |a| \ |b|$
\item $|a| \geq a$ and $|a| \geq -a$
\item $|a+b| \leq |a| + |b|$
\end{enumerate}


\section*{Question 6*}
Suppose $F$ is an ordered field and $a,b \in F$. Prove the following properties of the absolute value. 
\begin{enumerate}[(a)]
\item   $|b| \leq a$ if and only if $-a \leq b \leq a$;
\item   $||a|-|b|| \leq |a-b|$. 
\end{enumerate}
 \textbf{The second inequality is known as the reverse triangle inequality.}


\section*{Question 7*}
Suppose $F$ is an ordered field and $a,b, c \in F$. Prove the following properties of the absolute value. 
\textbf{We will use these inequalities repeatedly throughout the course.}
\begin{enumerate}[(a)]
\item Prove that $|a-b| \leq c$ if and only if $b-c \leq a \leq b+c$
\item Prove that $|a-b| < c$ if and only if $b-c < a < b+c$.
\end{enumerate}
 


\section*{Question 8}
Let $F$ be an ordered field, and let $0'$ and $1'$ denote the additive and multiplicative identities of $F$. For any $n \in \mathbb{N}$, let $n'$ denote the element of $F$ obtained by adding $1'$ to itself $n$ times; that is, 
\[ n' := \sum_{i=1}^n 1' = \underbrace{1' + \dots + 1'}_{n \text{ times}} . \]
Let $\mathbb{N}'$ denote the collection of  all such elements in $F$,
\[ \mathbb{N}':= \left\{ n' : n \in \mathbb{N}  \right\} \subseteq F   . \]
Define $f: \mathbb{N} \to \mathbb{N}'$ by $f(n) = n'$. Prove the following, for all $n,m, \in \mathbb{N}$:
\begin{enumerate}[(i)]
\item   $f$ is bijective;
\item $ f(n+m) = f(n) +f(m)$;
\item $f(n) < f(m) \iff n < m$;
\item $f(nm) = f(n)f(m)$.
\end{enumerate}

\section*{Question 9*}
This question is a continuation of the previous question. For any $S \subseteq F$, let $-S = \{ -s : s \in S \}$. Define
\[ \mathbb{Z}' := (-\mathbb{N}') \cup \{0' \} \cup \mathbb{N}' \subseteq F. \]
Define a function $g: \mathbb{Z} \to \mathbb{Z}'$ so that $g(n) = f(n)$ for all $n \in \mathbb{N}$, where $f$ is as in Question 8,  and $g$ satisfies properties (i)-(iv) from Question 8 for all $n, m \in \mathbb{Z}$. Justify your answer with a proof.

\section*{Question 10}
This question is a continuation of the previous questions. Define 
\[ \mathbb{Q}' =\left\{ \frac{p'}{q'} : q' \in \mathbb{Z}' \setminus \{0'\}, p' \in \mathbb{Z}' \right\} \subseteq F. \]
Define a function $h: \mathbb{Q} \to \mathbb{Q}'$ by
\[ h \left( \frac{p}{q} \right) = \frac{g(p)}{g(q)} , \text{ for all } q \in \mathbb{Z} \setminus \{0 \}, \ p \in \mathbb{Z} , \]
where $g$ is the function from the previous question. 
\begin{enumerate}[(i)]
\item Prove that $h$ is a well-defined function by showing that if $\frac{p}{q}= \frac{a}{b} \in \mathbb{Q}$ for $q,b \in \mathbb{Z} \setminus \{0\}$ and $p, a \in \mathbb{Z}$, then $h \left( \frac{p}{q} \right)= h \left( \frac{a}{b}\right)$.
\item Show that $h$ satisfies properties (i)-(iv) from question 5 for all $m,n \in \mathbb{Q}$.
\end{enumerate}
The function $h$ is an ordered field isomorphism from $\mathbb{Q}$ to $\mathbb{Q}' \subseteq F$. This shows that any ordered field $F$ has a subfield $\mathbb{Q}'$ that is isomorphic to $\mathbb{Q}$.

\label{VeryLastPage}
\end{document}

