Prove that for every polynomial $x^n$, $x^n(mod(x^4 + 1)) = x^{n(mod4)}$

Question

I am trying to prove the following:

Assuming $GF(2^k)[x]$ (where $k$ is a fixed natural number) is a ring of polynomials with coefficients in the field $GF(2^k)$. Prove that for every polynomial $x^n$ (where $n\in \mathbb{N}$) from $GF(2^k)[x]$ we have:

$$x^n(mod(x^4 + 1)) = x^{n(mod4)}$$

Any idea?

Thanks for your help.

Answer 1

for $a \gt 0$ $$ x^{4a+b} = (x^4+1)x^{4(a-1) +b}-x^{4(a-1) +b}\equiv x^{4(a-1) +b} $$ because characteristic is 2