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\noindent \textbf{Math 314 - Fall 2018 \hfill
Name:}
\hspace{2in} %Replace this line with your name!
\noindent \textbf{Mission 5} \hfill Due October 15th, 2018 \vspace{-4mm}\\
\small \textit{It used to be expensive to make things public and cheap to make them private. Now it’s expensive to make things private and cheap to make them public.}\\\vspace{1mm} \hfill--- Clay Shirky
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\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 Use the rules for Legendre symbols (not Jacobi symbols) and quadratic reciprocity to determine whether 41 is a square modulo 71.
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%Type your answer here!
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\item Repeat question 1 using the rules for Jacobi Symbols instead to determine whether 41 is a square modulo 71.
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%Type your answer here!
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\item Let $p \equiv 3 \pmod{4}$ be prime, and write $p = 4k + 3$. Give a proof that the equation $x^2 \equiv −1 \pmod{p}$ has no solutions. (Hint: Suppose $x$ exists. Raise both sides to the power $(p-1)/2$ and use
Fermat’s theorem.)
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%Type your answer here!
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\item Use Fermat's primality test to test 31 and 33 for primality using the bases 2 and 5.
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%Type your answer here!
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\item Use Euclid's algorithm to find the inverse of $f(x)=x^2$ in the field $\mathbb{F}_8$ with irreducible polynomial $x^3+x+1$.
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%Type your answer here!
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\end{enumerate}
\end{document}