GCD of polynomials in $\mathbb{F}_2[x]$

Question

How can one show that $\gcd(x^4+x+1,x^4-x)=1$ in $\mathbb{F}_2[x]$?

Is it because the polynomials do not share any irreducible monic polynomials i.e. $1, x, x+1$ as factors?

Answer 1

You have $(x^ 4+x+1)-(x^ 4-x)=2x+1=1$.

Answer 2

Don't forget $x^2+x+1$, $x^3+x+1$, $x^3+x^2+1$, $x^4 + x + 1$, $x^4 + x^3 + 1$, and $x^4 + x^3 + x^2 + x + 1$ are monic irreducible polynomials that could be relevant too!

But really, the easiest way in this case is just to compute the GCD with the Euclidean algorithm or some variant thereof.