\documentclass[12pt]{amsart}
\pagestyle{plain}
\usepackage{amsthm, setspace, framed, hyperref}
\usepackage{enumerate}
\usepackage{fullpage}
\begin{document}
\noindent \textbf{Math 314 - Spring 2019 \hfill
Name:}
\hspace{2in} %Replace this line with your name!
\noindent \textbf{Mission 5} \hfill Due March 12, 2019 \vspace{-4mm}\\
\small \textit{It used to be expensive to make things public and cheap to make them private. Now itâ€™s expensive to make things private and cheap to make them public.}\\\vspace{1mm} \hfill--- Clay Shirky
\vspace{-5mm}
\normalsize
\noindent \hrulefill
\section*{Guidelines}
\begin{itemize}
\item All work must be shown for full credit.
\item You can choose to use SageMath code to help you solve the problems. If you do, print out your code.
\item Either print out this assignment and write your answers on it, or edit the latex source. Make sure you still show your work!
There is one point of extra credit available on this assignment if you use \LaTeX
\item You may work with classmates, but be sure to turn in your own written solutions. Write down the name(s) of anyone who helps you.
\item Check one:\\
%You can put an x inside the framebox to "check" the box in latex for example: \framebox(12,12){x}
\framebox(12,12){} I worked with the following classmate(s):
\rule{7cm}{0.5pt}\\ %Replace this line with names of students.
\noindent\framebox(12,12){} I did not receive any help on this assignment.
\end{itemize}
\section{Graded Problems}
\begin{enumerate}[1.]
\item Let $f(x) = x^5+ x^3+x^2 +1$ and $g(x) = x^3 + x + 1$ be polynomials with coefficients in $\mathbb{F}_2$, the ring (field) of integers modulo 2. Compute $f(x) + g(x)$ and $f(x)\times g(x)$.
\begin{framed}
\vspace{3.8in}
%Type your answer here!
\end{framed}
\pagebreak
\item Write down all of the 8 elements of field $\mathbb{F}_8$ using the irreducible polynomial $x^3 + x +1$. Multiply each element by $x^2 + x$. What is the inverse of $x^2 +x$ in this field?
\begin{framed}
\vspace{3.5in}
%Type your answer here!
\end{framed}
\item Use the fermat primality test to test whether 33 is prime using first the base $a=10$ and then the base $a=2$.
\begin{framed}
\vspace{3.5in}
%Type your answer here!
\end{framed}
\item Use the rules for Legendre symbols (not Jacobi symbols) to determine whether 83 is a square modulo 149. (Note 149 is prime)
\begin{framed}
\vspace{3in}
%Type your answer here!
\end{framed}
\item Repeat question 1 using the rules for Jacobi Symbols instead to determine whether 83 is a square modulo 149. (Don't factor odd composite numbers)
\begin{framed}
\vspace{3in}
%Type your answer here!
\end{framed}
\end{enumerate}
\section{Recommended Exercises}
\noindent These will not be graded but are recommended if you need more practice.
\begin{itemize}
\item Section 3.13: \# 29, 33
\end{itemize}
\end{document}