Describing all $f \in \mathbb{F}_2[x]$ divisible by $x^2 +1$
Solution 1:
First, show that if $f(x)=f_0+f_1x+\dots+f_nx^n$, then $$ f(x)\equiv (f_0+f_2+\dots)+x(f_1+f_3+\dots)\pmod{x^2+1} $$ Conclude that $f(x)$ is a multiple of $x^2+1$ if and only if the sum of the even index coefficients is $0$, as well as the sum of the odd-index coefficients.