$[$Q$-2$ Total $8$ Marks$]$

$p(x)=x^6+x+1$ is claimed to be an irreducible polynomial $p(x)$ for $GF(2^6)\frac{x}{p}(x)($See Page $25$ paragraph $2$ of class notes$).$ Show the steps to verify this claim. $($Hint: consider using the deconv function in Octave$/$Matlab and document in your answer book how you used this function to verify this claim, if you use it$.)$

Compute the element corresponding to $(x^3+x+1{)}^7$ in $GF(2^6)\frac{x}{p}(x)$ with $p(x)=x^6+x+1.($Hint: The Octave$/$Matlab functions conv, deconv could be used for computing this. Document how you use these functions so I can verify your computations.$)$