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\section{Solutions and hints for exercises}  \small{  \textbf{Exercise 1.}  \begin{itemize}  \item $(1+1)+0=(0)+0=0$; 
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\end{itemize}  \textbf{Exercise 18}  The only irreducible polynomial of $\Fb[x]$ with degree $2$ is $x^2+x+1$; now since it does not divide $f=x^4+x+1$ and $f$ has no roots in $\Fb$, we can desume that it is irreducible.  }