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\begin{center} {\large \textbf{Math 437 -- Homework 3}} \smallskip \\ 
Due 10:15am on Thursday, February 2, 2017  \\ \ \\
 \end{center}

Please indicate any sources you used for a given problem on the solution to that problem. 
For example, if you worked with another student to get the solution to a problem, please indicate who.
You are welcome to work together in small groups, but please try the problems on your own first. \\ \ \\



\begin{prob}
Let $a = x^6 + x^2 + 1$ and $b = x^4 + x^3 + x^2 +x + 1$ in $\Z_2[x]$. 
\begin{itemize} 
\item[(a)] Find the greatest common divisor of $a$ and $b$. 
\item[(b)] Find polynomials $c, d \in \Z_2[x]$ so that $ac + bd = \gcd(a,b)$. 
  \end{itemize} 
  \end{prob}\ \\

\begin{prob} \
\begin{itemize} 
\item[(a)] Let $F$ be a field and $f(x) \in F[x]$ be a polynomial of degree $2$ or $3$. Show that $f$ is irreducible if and only if 
$f(x)$ does not have any roots in $F$
\item[(b)] Find all irreducible polynomials of degree 1, 2, and 3 in $\Z_2[x]$.
\item[(c)] Give an example of a reducible polynomial of degree $4$ in $\Z_2[x]$ with no roots in $\Z_2$. 
  \end{itemize}
\end{prob} \ \\


\begin{prob} Let $F$ be the field $\Z_2[x]/(f)$ where $f = x^3 + x^2 +1$. 
\begin{itemize} 
\item[(a)] How many elements are in $F$ and $F^*$? 
\item[(b)] For each $1\leq k \leq 7$, find a polynomial of degree $\leq 2$ in $\Z_2[x]$ that equals $x^k$ mod  $(f)$.  
Is $f$ primitive? 
\item[(c)] Find a multiplicative inverse for $x^2 + x+ 1$ in $F$. 
  \end{itemize}
\end{prob} \ \\




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