\begin
Discover why over 20 million people worldwide trust Overleaf with their work.
\documentclass[11pt,letterpaper]{article}
\usepackage[lmargin=1in,rmargin=1in,tmargin=1in,bmargin=1in]{geometry}
% -------------------
% Packages
% -------------------
\usepackage{
amsmath, % Math Environments
amssymb, % Extended Symbols
amsthm, % Theorem Environments
cancel, % Use Cancels
enumerate, % Enumerate Environments
graphicx, % Include Images
lastpage, % Reference Lastpage
multicol, % Use Multi-columns
multirow, % Use Multi-rows
xcolor % Use Colors
}
% -------------------
% Font
% -------------------
\usepackage[T1]{fontenc}
\usepackage{charter}
%\usepackage[T1]{fontenc}
%\usepackage{mathpazo}
%\usepackage[bitstream-charter]{mathdesign}
%\usepackage[T1]{fontenc}
% -------------------
% Tikz & PGF
% -------------------
\usepackage{tikz}
\usepackage{tikz-cd}
\usetikzlibrary{
calc,
decorations.pathmorphing,
matrix,arrows,
positioning,
shapes.geometric
}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
% -------------------
% Commands
% -------------------
% Problem Labels
\newcounter{problem}
\newcommand{\problem}{
\stepcounter{problem} %
\noindent \textbf{Problem \theproblem. }}
\newcommand{\solution}{\noindent \textbf{Solution: }}
\newcommand{\pf}{\noindent\emph{Proof. }}
% Special Characters
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
% Math Operators
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\disc}{disc}
\DeclareMathOperator{\Span}{span}
% Special Commands
\newcommand{\pspace}{\par\vspace{\baselineskip}}
\newcommand{\ds}{\displaystyle}
\newcommand{\defeq}{\stackrel{\text{def}}{=}}
\newcommand{\ov}[1]{\overline{#1}}
\newcommand{\ma}[1]{\stackrel{#1}{\longrightarrow}}
\newcommand{\twomatrix}[4]{\begin{pmatrix} #1 & #2 \\ #3 & #4 \end{pmatrix}}
% -------------------
% Header & Footer
% -------------------
\usepackage{fancyhdr}
\fancypagestyle{title}{
%Headers
\fancyhead[L]{\Large My Name}
\fancyhead[C]{\Large MAT 999}
\fancyhead[R]{\Large Homework \#}
\renewcommand{\headrulewidth}{0.2pt}
%Footers
\fancyfoot[L]{}
\fancyfoot[C]{}
\fancyfoot[R]{\thepage \,of \pageref{LastPage}}
\renewcommand{\footrulewidth}{0.0pt}
}
\fancypagestyle{pages}{
%Headers
\fancyhead[L]{}
\fancyhead[C]{}
\fancyhead[R]{}
\renewcommand{\headrulewidth}{0.0pt}
%Footers
\fancyfoot[L]{}
\fancyfoot[C]{}
\fancyfoot[R]{\thepage \,of \pageref{LastPage}}
\renewcommand{\footrulewidth}{0.0pt}
}
\headheight=18pt
\footskip=14pt
\pagestyle{pages}
% -------------------
% Content
% -------------------
\begin{document}
\thispagestyle{title}
% Problem 1
\problem Show that there exists no nontrivial unramified extensions of $\Q$. \pspace
\solution If $K/\Q$ is a nontrivial number field, then $|\disc K|>1$. But then $\disc K$ has a prime factor so that some prime ramifies in $K$. \qed \pspace
% Problem 2
\problem Complete the following:
\begin{enumerate}[(a)]
\item How does one prove a cotheorem?
\item Compute $\ds \int \cos x \;dx$.
\item How does one square $\twomatrix{a}{b}{c}{d}$?
\end{enumerate}
\solution
\begin{enumerate}[(a)]
\item Use rollaries.
\item We have
\begin{equation} \label{eq:integral}
\int \cos x \;dx= \sin x + C
\end{equation}
We can check \eqref{eq:integral}:
\[
\dfrac{d}{dx} \left( \sin x + C \right)= \cos x
\]
\item This is routine.
\end{enumerate} \qed \pspace
% Problem 3
\problem Prove that $\sqrt{2}$ is irrational. \pspace
\pf Assume that $\sqrt{2}= \dfrac{a}{b}$, where $a,b \in \Z$. Without loss of generality, we may assume $\gcd(a,b)= 1$. Then we have
\begin{align}
\sqrt{2}&= \dfrac{a}{b} \nonumber \\
\sqrt{2}^2&= \left( \dfrac{a}{b} \right)^2 \label{eq:implication1} \\
2&= \dfrac{a^2}{b^2} \nonumber \\
a^2&= 2b^2 \label{eq:implication2}
\end{align}
But then from \eqref{eq:implication2}, we know that $a^2$ is even so that $a$ is even. But then we must have
\[
2a^2= b^2
\]
so that $b^2$ is even, implying $b$ is even. But then $\gcd(a,b) \geq 2$, a contradiction. \qed \\
\end{document}