How could i prove this?
Let $F$ be a finite field of characteristic $2$ and $g \in F[X]$ an irreducible polynomial. Splitting this polynomial in even and odd part we get $g(X)=g_0(X)^2+Xg_1(X)^2$. Then there exists a polynomial $w \in F[X]$ such that $w^2(X)\equiv X \bmod g(X)$.