Need to factorize$ x^7 - x$ to irreducible polynomials over $\operatorname{GF}(2)$

Question

Can some one help me with an easy method to solve this question?

Factorize $x^7 - x$ to irreducible polynomials over $\operatorname{GF}(2)$

Answer 1

Using that modulo $\;2\;$ we have $\;(a+b)^2=a^2+b^2\;$ :

$$x^7-x=x(x^6-1)=x(x^3-1)^2=x(x-1)^2(x^2+x+1)^2$$

Of course, $\;-1=1\;$ so you can write $\;x-1\;$ or $\;x+1\;$ . It's the same.