Find ged a$($x$),$b$($x$)]$ for a$($x$)=$x$+$x$+$x$+$x$+$x$^2+$x$+1$ and b$($x$)=$x$+$x$^2+$x$+1.$

Prove that:

i$.$ Addition operation in GF$(2)$ is equivalent to exclusive$-$OR $($XOR$)$ operation.

ii$.$ Multiplication operation in GF $(2)$ is equivalent to logical AND operation.