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\usepackage{enumitem}
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\title{\vspace{-5em}Coding and Cryptography Fall 2016\\ Worksheet on finite fields}
\author{Katherine E. Stange}
\date{\today}
\begin{document}
\maketitle %this is what makes the title from the data above
We will study the set $\mathbb{F}_4$ of polynomials in $X$ with coefficients modulo $2$, considered modulo $X^2 + X + 1$.
\begin{enumerate}
\item For example, in this world,
\[
X^2 = X+1.
\]
Explain why.
\vspace{1em}
\item List all the elements of $\mathbb{F}_4$.
\vspace{10em}
\item Determine the full addition and multiplication tables of $\mathbb{F}_4$.
\newpage
\item Write a substraction and division table for $\mathbb{F}_4$. You can put ``not defined'' when division is not possible.
\vspace{19em}
\item By looking at the tables you've created, verify whether or not the following properties hold:
\begin{enumerate}
\item There is an element $e_A$ which satisfies $a+e_A = a$ for all $a$. What is it?
\vspace{1em}
\item There is an element $e_M$ which satsifies $ae_M = a$ for all $a$. What is it?
\vspace{1em}
\item The elements $e_A$ and $e_M$ are distinct.
\vspace{1em}
\item For each $a$, there is an element $a'_A$ satisfying $a + a'_A = e_A$. Explain.
\vspace{1em}
\item For each non-zero $a$, there is an element $a'_M$ satisfying $a a'_M = e_M$. Explain.
\vspace{1em}
\item Addition is associative (give an example of this property).
\vspace{1em}
\item Multiplication is associative (give an example of this property).
\vspace{1em}
\item Addition is commutative (give an example of this property).
\vspace{1em}
\item Multiplication is commutative (give an example of this property).
\vspace{1em}
\item Multiplication distributes over addition (give an example of this property).
\end{enumerate}
\vspace{1em}
\item The fact that $\mathbb{F}_4$ satisfies the above axioms shows it is a field. It is a field of four elements. Can you construct a field of 9 elements? What sizes are possible by this method?
\end{document}